{"id":1017,"date":"2020-08-07T19:04:04","date_gmt":"2020-08-07T19:04:04","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/understand-slope-of-a-line-2\/"},"modified":"2025-03-25T18:39:39","modified_gmt":"2025-03-25T18:39:39","slug":"understand-slope-of-a-line-2","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/understand-slope-of-a-line-2\/","title":{"raw":"3.4 Understand Slope of a Line","rendered":"3.4 Understand Slope of a Line"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section it is expected that you will be able to:\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li>Use \\(m=\\dfrac{rise}{run}\\) to find the slope of a line from its graph<\/li>\r\n \t<li>Find the slope of horizontal and vertical lines<\/li>\r\n \t<li>Use the slope formula to find the slope of a line between two points<\/li>\r\n \t<li>Graph a line given a point and the slope<\/li>\r\n \t<li>Solve slope applications<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169597691001\">When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?<\/p>\r\n<p id=\"fs-id1169597447394\">In mathematics, the \u2018tilt\u2019 of a line is called the <em data-effect=\"italics\">slope<\/em> of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.<\/p>\r\n<p id=\"fs-id1169597704220\">In this section, we will explore the concept of slope.<\/p>\r\n<p id=\"fs-id1169597618132\">The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter \\(m\\).<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Slope of a line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1169597569399\">The <span data-type=\"term\">slope of a line<\/span> of a line is \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\).<\/p>\r\n<p id=\"fs-id1169597775407\">The <span data-type=\"term\">rise<\/span> measures the vertical change and the <span data-type=\"term\">run<\/span> measures the horizontal change between two points on the line.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Positive and negative slopes<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1169597508120\">We \u2018read\u2019 a line from left to right just like we read words in English. As you read from left to right, the line\u00a0 is going up; it has <span data-type=\"term\">positive slope<\/span>. The line is going down; it has <span data-type=\"term\">negative slope<\/span>.<\/p>\r\n\r\n<div id=\"fs-id1169595286254\" data-type=\"note\">\r\n<div data-type=\"title\"><\/div>\r\n<span id=\"fs-id1169597576838\" data-type=\"media\" data-alt=\"The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled \u201cPositive slope\u201d. The line on the right is a diagonal line that drops from left to right. It is labeled \u201cNegative slope\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2020\/08\/CNX_ElemAlg_Figure_04_04_043_img_new.jpg\" alt=\"The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled \u201cPositive slope\u201d. The line on the right is a diagonal line that drops from left to right. It is labeled \u201cNegative slope\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Use \\(m=\\dfrac{\\text{rise}}{\\text{run}}}\\) to Find the Slope of a Line from its Graph<\/h1>\r\n<p id=\"fs-id1169597708542\">We\u2019ll look at some graphs on the \\(xy\\)-coordinate plane and see how to find their slopes.<\/p>\r\n<p id=\"fs-id1169595180855\">To find the slope, we must count out the rise and the run. But where do we start?<\/p>\r\n<p id=\"fs-id1169597483663\">We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div data-type=\"title\">How to Use \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\) to Find the Slope of a Line from its Graph<\/div>\r\n<div id=\"fs-id1169595179156\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597514142\" data-type=\"problem\">\r\n<p id=\"fs-id1169595237681\">Find the slope of the line shown.<\/p>\r\n<span id=\"fs-id1169597533738\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_013_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597807891\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<span id=\"fs-id1169597422743\" data-type=\"media\" data-alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).\u201d To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061a_img_new.jpg\" alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).\u201d To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597740024\" data-type=\"media\" data-alt=\"The second row says, \u201cStep 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).\u201d In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061b_img_new.jpg\" alt=\"The second row says, \u201cStep 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).\u201d In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597536602\" data-type=\"media\" data-alt=\"The third row then says, \u201cStep 3. Count the rise and the run on the legs of the triangle.\u201d The rise is 4 and the run is 5.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061c_img_new.jpg\" alt=\"The third row then says, \u201cStep 3. Count the rise and the run on the legs of the triangle.\u201d The rise is 4 and the run is 5.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597422236\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.\u201d To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.\u201d To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595313202\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597837782\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\r\n<p id=\"fs-id1169597467540\">Find the slope of the line shown.<\/p>\r\n<span id=\"fs-id1169597524912\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_048_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595663978\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169597825152\">\\(\\dfrac{2}{5}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Find the slope of a line from its graph using m = rise \/ run.<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1169597712904\" class=\"stepwise\" type=\"1\">\r\n \t<li>Locate two points on the line whose coordinates are integers.<\/li>\r\n \t<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\r\n \t<li>Count the rise and the run on the legs of the triangle.<\/li>\r\n \t<li>Take the ratio of rise to run to find the slope, \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\).<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595313202\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597837782\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the slope of the line shown.\r\n<div id=\"fs-id1169595217562\" data-type=\"problem\">\r\n\r\n<span id=\"fs-id1169597721249\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_017_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595310501\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"fs-id1168465017968\" style=\"width: 100%;\" summary=\"This figure shows step-by-step how to find the slope of the line with points (0, 5) and (3, 3). First, identify the leftmost point, which is (0, 5). Starting at (0, 5), sketch a right triangle to (3, 3). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 9. The y-axis of the plane runs from negative 1 to 7. The points (0, 5) and (3, 3) are plotted. An additional point is plotted at (0, 3). The three points form a right triangle, with the line from (0, 5) to (3, 3) forming the hypotenuse and the lines from (0, 5) to (0, 3) and from (0, 3) to (3, 3) forming the legs. The leg from (0, 5) to (0, 3) is labeled \u201crise\u201d and the leg from (0, 3) to (3, 3) is labeled \u201crun\u201d. The next step is to count the rise, which is negative. The rise is negative 2. The next step is to count the run, which is 3. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals negative 2 thirds.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Locate two points on the graph whose coordinates are integers.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,5\\right)\\) and \\(\\left(3,3\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Which point is on the left?<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,5\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Starting at \\(\\left(0,5\\right)\\), sketch a right triangle to \\(\\left(3,3\\right)\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169595186611\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_018a_img_new.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Count the rise\u2014it is negative.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The rise is \\(-2\\).<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Count the run.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The run is 3.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the slope formula.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(m=\\dfrac{\\text{rise}}{\\text{run}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of the rise and run.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(m=\\dfrac{-2}{3}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(m=-\\dfrac{2}{3}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The slope of the line is \\(-\\dfrac{2}{3}\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169597467837\">So \\(y\\) increases by 3 units as \\(x\\) decreases by 2 units.<\/p>\r\n<p id=\"fs-id1169597701404\">What if we used the points \\(\\left(-3,7\\right)\\) and \\(\\left(6,1\\right)\\) to find the slope of the line?<\/p>\r\n<span id=\"fs-id1169595297895\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_029_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169595305929\">The rise would be \\(-6\\) and the run would be 9. Then \\(m=\\dfrac{-6}{9}\\), and that simplifies to \\(m=-\\dfrac{2}{3}\\). Remember, it does not matter which points you use\u2014the slope of the line is always the same.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597876171\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595339399\" data-type=\"exercise\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1169595180581\">Find the slope of the line shown.<\/p>\r\n<span id=\"fs-id1169595273647\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_050_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597602160\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169595195547\">\\(-\\dfrac{4}{3}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169597536164\">In the last two examples, the lines had <em data-effect=\"italics\">y<\/em>-intercepts with integer values, so it was convenient to use the <em data-effect=\"italics\">y<\/em>-intercept as one of the points to find the slope. In the next example, the <em data-effect=\"italics\">y<\/em>-intercept is a fraction. Instead of using that point, we\u2019ll look for two other points whose coordinates are integers. This will make the slope calculations easier.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595185825\" data-type=\"problem\">\r\n<p id=\"fs-id1169597555840\">Find the slope of the line shown.<\/p>\r\n<span id=\"fs-id1169597555844\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_020_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597374087\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"fs-id1168464877438\" style=\"width: 100%;\" summary=\"This figure shows step-by-step how to find the slope of the line with points (2, 3) and (7, 6). First, identify the leftmost point, which is (2, 3). Starting at (2, 3), sketch a right triangle to (7, 6). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis of the plane runs from 0 to 7. The points (2, 3) and (7, 6) are plotted. An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The leg from (2, 3) to (2, 6) is labeled \u201crise\u201d and the leg from (2, 6) to (7, 6) is labeled \u201crun\u201d. The next step is to count the rise, which is 3. The next step is to count the run, which is 5. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals 3 fifths.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Locate two points on the graph whose coordinates are integers.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(2,3\\right)\\) and \\(\\left(7,6\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Which point is on the left?<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(2,3\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Starting at \\(\\left(2,3\\right)\\), sketch a right triangle to \\(\\left(7,6\\right)\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169595227774\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_021a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Count the rise.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The rise is 3.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Count the run.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The run is 5.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the slope formula.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(m=\\dfrac{\\text{rise}}{\\text{run}}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of the rise and run.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(m=\\dfrac{3}{5}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The slope of the line is \\(\\dfrac{3}{5}\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169595287958\">This means that \\(y\\) increases 5 units as \\(x\\) increases 3 units.<\/p>\r\n<p id=\"fs-id1169597838861\">When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.<\/p>\r\n<p id=\"fs-id1169595312791\">What would happen if we started with the point on the right?<\/p>\r\n<p id=\"fs-id1169597705426\">Let\u2019s use the points \\(\\left(2,3\\right)\\) and \\(\\left(7,6\\right)\\) again, but now we\u2019ll start at \\(\\left(7,6\\right)\\).<\/p>\r\n<span id=\"fs-id1169595362196\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_022_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<table id=\"eip-950\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Count the rise.<\/td>\r\n<td>The rise is \\(-3\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run. It goes from right to left, so it is negative.<\/td>\r\n<td>The run is \\(-5\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>\\(m=\\dfrac{\\text{rise}}{\\text{run}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>\\(m=\\dfrac{-3}{-5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The slope of the line is \\(\\dfrac{-3}{-5}\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169595219335\">It does not matter where you start\u2014the slope of the line is always the same.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595120989\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597701127\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597689410\" data-type=\"problem\">\r\n<p id=\"fs-id1169597689412\">Find the slope of the line shown.<\/p>\r\n<span data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_052_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595257080\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169597537337\">\\(\\dfrac{5}{4}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Find the Slope of Horizontal and Vertical Lines<\/h1>\r\n<p id=\"fs-id1169597682080\">Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.<\/p>\r\n\\(\\begin{array}{cc}\\textbf{Horizontal line} \\quad \\mathbf{y=b}\\hfill &amp; \\textbf{Vertical line}\\quad \\mathbf{x=a}\\hfill \\\\ \\\\ \\text{y-coordinates are the same.}\\hfill &amp; \\text{x-coordinates are the same.}\\hfill \\end{array}\\)\r\n<p id=\"fs-id1169595318009\">So how do we find the slope of the horizontal line \\(y=4\\)? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let\u2019s see what happens when we do this.<\/p>\r\n<span id=\"fs-id1169597456359\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_023_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).\" data-media-type=\"image\/jpeg\"><\/span>\r\n<table id=\"eip-321\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>What is the rise?<\/td>\r\n<td>The rise is \\(0\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is \\(3\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the slope?<\/td>\r\n<td>\\(\\begin{array}{l}m=\\dfrac{\\text{rise}}{\\text{run}}\\\\ m=\\dfrac{0}{3}\\\\ m=0\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The slope of the horizontal line \\(y=4\\) is \\(0\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169597725101\">All horizontal lines have slope 0. When the <em data-effect=\"italics\">y<\/em>-coordinates are the same, the rise is 0.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Slope of a horizontal line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe slope of a horizontal line, \\(y=b\\), is 0.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169597468033\">The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.<\/p>\r\nNow, we\u2019ll consider a vertical line, the line.\r\n\r\n<span id=\"fs-id1169595174471\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_024_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).\" data-media-type=\"image\/jpeg\"><\/span>\r\n<table id=\"eip-315\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>What is the rise?<\/td>\r\n<td>The rise is \\(2\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is \\(0\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the slope?<\/td>\r\n<td>\\(\\begin{array}{l} m=\\dfrac{\\text{rise}}{\\text{run}}\\\\ m=\\dfrac{2}{0}\\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169595663765\">But we can\u2019t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line \\(x=3\\) is undefined.<\/p>\r\n<p id=\"fs-id1169595226611\">The slope of any vertical line is undefined. When the <em data-effect=\"italics\">x<\/em>-coordinates of a line are all the same, the run is 0.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Slope of a vertical line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe slope of a vertical line, \\(x=a\\), is undefined.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595195537\" data-type=\"note\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597714862\" data-type=\"problem\">\r\n<p id=\"fs-id1169597714865\">Find the slope of each line:<\/p>\r\n<p id=\"fs-id1169597714866\"><span class=\"token\">a) <\/span>\\(x=8\\)\u2003b) \\(y=-5\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597739770\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1168465232289\"><span class=\"token\">a) <\/span>\\(x=8\\)<span data-type=\"newline\">\r\n<\/span> This is a vertical line.<span data-type=\"newline\">\r\n<\/span> Its slope is undefined.<span data-type=\"newline\">\r\n<\/span><span data-type=\"newline\">\r\n<\/span><span class=\"token\">b)<\/span> \\(y=-5\\)<span data-type=\"newline\">\r\n<\/span> This is a horizontal line.<span data-type=\"newline\">\r\n<\/span> It has slope 0.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597465640\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597465644\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597413629\" data-type=\"problem\">\r\n<p id=\"fs-id1169597413631\">Find the slope of the line: \\(x=-4.\\)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597507381\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169597507383\">undefined<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595176638\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595118607\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597818064\" data-type=\"solution\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Quick guide to the slopes of lines<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597517831\" data-type=\"note\">\r\n<div data-type=\"title\"><\/div>\r\n<span id=\"fs-id1169597761828\" data-type=\"media\" data-alt=\"This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_054_img_new.jpg\" alt=\"This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<p id=\"fs-id1169597877171\">Remember, we \u2018read\u2019 a line from left to right, just like we read written words in English.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Use the Slope Formula to find the Slope of a Line Between Two Points<\/h1>\r\n<p id=\"fs-id1169597594129\">Sometimes we\u2019ll need to find the slope of a line between two points when we don\u2019t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we\u2019ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.<\/p>\r\n<p id=\"fs-id1169597767915\">We have seen that an ordered pair \\(\\left(x,y\\right)\\) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol \\(\\left(x,y\\right)\\) be used to represent two different points? Mathematicians use subscripts to distinguish the points.<\/p>\r\n\\(\\begin{array}{cc}\\left({x}_{1},{y}_{1}\\right)\\hfill &amp;\u00a0 \\text{read \u2018}\\enspace x \\enspace \\text{sub 1,} \\enspace y \\enspace \\text{sub 1\u2019}\\hfill \\\\ \\left({x}_{2},{y}_{2}\\right)\\hfill &amp; \\text{read \u2018} \\enspace x \\enspace \\text{sub 2,}\\enspace\u00a0 y \\enspace \\text{sub 2\u2019}\\hfill \\end{array}\\)\r\n<p id=\"fs-id1169595174195\">The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?<\/p>\r\n<p id=\"fs-id1169595219379\">We will use \\(\\left({x}_{1},{y}_{1}\\right)\\) to identify the first point and \\(\\left({x}_{2},{y}_{2}\\right)\\) to identify the second point.<\/p>\r\n<p id=\"fs-id1169597700360\">If we had more than two points, we could use \\(\\left({x}_{3},{y}_{3}\\right)\\), \\(\\left({x}_{4},{y}_{4}\\right)\\), and so on.<\/p>\r\n<p id=\"fs-id1169595147483\">Let\u2019s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points \\(\\left(2,3\\right)\\) and \\(\\left(7,6\\right)\\).<\/p>\r\n<span id=\"fs-id1169597494106\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_025_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169597618262\">Since we have two points, we will use subscript notation, \\(\\begin{pmatrix}{x}_{1},&amp;{y}_{1}\\\\ 2, &amp; 3 \\end{pmatrix}\\)\\(\\begin{pmatrix}{x}_{2}, &amp; {y}_{2} \\\\ 7, &amp; 6\\end{pmatrix}\\).<\/p>\r\n<p id=\"fs-id1169597689516\">On the graph, we counted the rise of 3 and the run of 5.<\/p>\r\n<p id=\"fs-id1169595122971\">Notice that the rise of 3 can be found by subtracting the <em data-effect=\"italics\">y<\/em>-coordinates 6 and 3.<\/p>\r\n\\(3=6-3\\)\r\n<p id=\"fs-id1169595275565\">And the run of 5 can be found by subtracting the <em data-effect=\"italics\">x<\/em>-coordinates 7 and 2.<\/p>\r\n\\(5=7-2\\)\r\n<p id=\"fs-id1169595354853\">We know \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\). So \\(m=\\dfrac{3}{5}\\).<\/p>\r\n<p id=\"fs-id1169595344242\">We rewrite the rise and run by putting in the coordinates \\(m=\\dfrac{6-3}{7-2}\\).<\/p>\r\n<p id=\"fs-id1169595119525\">But 6 is \\({y}_{2}\\), the <em data-effect=\"italics\">y<\/em>-coordinate of the second point and 3 is \\({y}_{1}\\), the <em data-effect=\"italics\">y<\/em>-coordinate of the first point.<\/p>\r\n<p id=\"fs-id1169597723209\">So we can rewrite the slope using subscript notation. \\(m=\\dfrac{{y}_{2}-{y}_{1}}{7-2}\\)<\/p>\r\n<p id=\"fs-id1169595311347\">Also, 7 is \\({x}_{2}\\), the <em data-effect=\"italics\">x<\/em>-coordinate of the second point and 2 is \\({x}_{1}\\), the <em data-effect=\"italics\">x<\/em>-coordinate of the first point.<\/p>\r\n<p id=\"fs-id1169595213803\">So, again, we rewrite the slope using subscript notation. \\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\)<\/p>\r\n<p id=\"fs-id1169595308709\">We\u2019ve shown that \\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\) is really another version of \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\). We can use this formula to find the slope of a line when we have two points on the line.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Slope formula<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1169597507992\">The slope of the line between two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is<\/p>\r\n\\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\)\r\n<p id=\"fs-id1169595287765\">This is the <span data-type=\"term\">slope formula<\/span>.<\/p>\r\n<p id=\"fs-id1169595255399\">The slope is:<\/p>\r\n\\(\\begin{array}{c}\\\\\u00a0 \u00a0\\text{y of the second point minus y of the first point}\\hfill \\\\ \\hfill \\text{over}\\hfill \\\\ \\hfill \\text{x of the second point minus x of the first point.}\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169597538694\" data-type=\"note\">\r\n<div id=\"fs-id1169595255402\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595125760\" data-type=\"problem\">\r\n<p id=\"fs-id1169597721158\">Use the <span class=\"no-emphasis\" data-type=\"term\">slope formula<\/span> to find the slope of the line between the points \\(\\left(1,2\\right)\\) and \\(\\left(4,5\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597537093\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-758\" style=\"height: 160px; width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 32px;\">\r\n<td style=\"height: 32px; width: 303.438px;\">We'll call \\(\\left(1,2\\right)\\) point #1 and \\(\\left(4,5\\right)\\) point #2.<\/td>\r\n<td style=\"height: 32px; width: 546.781px;\">\\(\\begin{pmatrix}{x}_{1},&amp;{y}_{1}\\\\ 1, &amp; 2 \\end{pmatrix}\\)\\(\\begin{pmatrix}{x}_{2}, &amp; {y}_{2} \\\\ 4, &amp; 5\\end{pmatrix}\\).<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 303.438px;\">Use the slope formula.<\/td>\r\n<td style=\"height: 16px; width: 546.781px;\">\\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\).<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 303.438px;\">Substitute the values.<\/td>\r\n<td style=\"height: 16px; width: 546.781px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 32px;\">\r\n<td style=\"height: 32px; width: 303.438px;\">\\(y\\) of the second point minus \\(y\\) of the first point<\/td>\r\n<td style=\"height: 32px; width: 546.781px;\">\\(m=\\dfrac{5-2}{{x}_{2}-{x}_{1}}\\).<\/td>\r\n<\/tr>\r\n<tr style=\"height: 32px;\">\r\n<td style=\"height: 32px; width: 303.438px;\">\\(x\\) of the second point minus \\(x\\) of the first point<\/td>\r\n<td style=\"height: 32px; width: 546.781px;\">\\(m=\\dfrac{5-2}{4-1}\\).<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 303.438px;\">Simplify the numerator and the denominator.<\/td>\r\n<td style=\"height: 16px; width: 546.781px;\">\\(m=\\dfrac{3}{3}\\).<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 303.438px;\">Simplify.<\/td>\r\n<td style=\"height: 16px; width: 546.781px;\">\\(m=1\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169595104074\">Let\u2019s confirm this by counting out the slope on a graph using \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\).<\/p>\r\n<span id=\"fs-id1169597690980\" data-type=\"media\" data-alt=\"The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled \u201crise\u201d and the leg from (1, 5) to (4, 5) is labeled \u201crun\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_026_img_new.jpg\" alt=\"The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled \u201crise\u201d and the leg from (1, 5) to (4, 5) is labeled \u201crun\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169595176270\">It doesn\u2019t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595362728\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597805559\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597805561\" data-type=\"problem\">\r\n<p id=\"fs-id1169597805563\">Use the slope formula to find the slope of the line through the points: \\(\\left(8,5\\right)\\) and \\(\\left(6,3\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595123179\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169595123181\">1<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595309361\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597836954\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597603850\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597740105\" data-type=\"problem\">\r\n<p id=\"fs-id1169597740107\">Use the slope formula to find the slope of the line through the points \\(\\left(-2,-3\\right)\\) and \\(\\left(-7,4\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597807761\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-471\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>We'll call \\(\\left(-2,-3\\right)\\) point #1 and \\(\\left(-7,4\\right)\\) point #2.<\/td>\r\n<td>\\(\\begin{pmatrix}{x}_{1},&amp;{y}_{1}\\\\ -2, &amp; -3 \\end{pmatrix}\\)\\(\\begin{pmatrix}{x}_{2}, &amp; {y}_{2} \\\\ -7, &amp; 4\\end{pmatrix}\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>\\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\\(y\\) of the second point minus \\(y\\) of the first point<\/td>\r\n<td>\\(m=\\dfrac{4-\\left(-3\\right)}{{x}_{2}-{x}_{1}}\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\\(x\\) of the second point minus \\(x\\) of the first point<\/td>\r\n<td>\\(m=\\dfrac{4-\\left(-3\\right)}{-7-\\left(-2\\right)}\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(\\begin{array}{c}m=\\dfrac{7}{-5}\\hfill \\\\ m=-\\dfrac{7}{5}\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169597814147\">Let\u2019s verify this slope on the graph shown.<\/p>\r\n<span id=\"fs-id1169595106906\" data-type=\"media\" data-alt=\"The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled \u201crise\u201d and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled \u201crun\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_027_img_new.jpg\" alt=\"The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled \u201crise\u201d and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled \u201crun\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<div id=\"fs-id1169597824078\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill m&amp; =\\hfill &amp; \\dfrac{\\text{rise}}{\\text{run}}\\hfill \\\\ \\hfill m&amp; =\\hfill &amp; \\dfrac{-7}{5}\\hfill \\\\ \\hfill m&amp; =\\hfill &amp; -\\dfrac{7}{5}\\hfill \\end{array}\\)<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595228085\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595228089\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597508758\" data-type=\"problem\">\r\n<p id=\"fs-id1169597508760\">Use the slope formula to find the slope of the line through the points: \\(\\left(-3,4\\right)\\) and \\(\\left(2,-1\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597821110\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169597821112\">\\(-1\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Graph a Line Given a Point and the Slope<\/h1>\r\n<p id=\"fs-id1169597413558\">Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.<\/p>\r\n<p id=\"fs-id1169597421283\">One other method we can use to graph lines is called the <span data-type=\"term\">point\u2013slope method<\/span>. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div data-type=\"title\">How To Graph a Line Given a Point and The Slope<\/div>\r\n<div id=\"fs-id1169595256096\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597466646\" data-type=\"problem\">\r\n<p id=\"fs-id1169597773540\">Graph the line passing through the point \\(\\left(1,-1\\right)\\) whose slope is \\(m=\\dfrac{3}{4}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595176366\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<span id=\"fs-id1169597715877\" data-type=\"media\" data-alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Plot the given point. Plot (1, negative 1).\u201d To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062a_img_new.jpg\" alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Plot the given point. Plot (1, negative 1).\u201d To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597430242\" data-type=\"media\" data-alt=\"The second row says, \u201cStep 2. Use the slope formula m equals rise divided by run to identify the rise and the run.\u201d The rise and run are 3 and 4, so m equals 3 divided by 4.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062b_img_new.jpg\" alt=\"The second row says, \u201cStep 2. Use the slope formula m equals rise divided by run to identify the rise and the run.\u201d The rise and run are 3 and 4, so m equals 3 divided by 4.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597521254\" data-type=\"media\" data-alt=\"The third row says \u201cStep 3. Starting at the given point, count out the rise and run to mark the second point.\u201d We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062c_img_new.jpg\" alt=\"The third row says \u201cStep 3. Starting at the given point, count out the rise and run to mark the second point.\u201d We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1169597331408\" data-type=\"media\" data-alt=\"The fourth row says \u201cStep 4. Connect the points with a line.\u201d On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062d_img_new.jpg\" alt=\"The fourth row says \u201cStep 4. Connect the points with a line.\u201d On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595211024\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595211029\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595340096\" data-type=\"problem\">\r\n<p id=\"fs-id1169595340098\">Graph the line passing through the point \\(\\left(2,-2\\right)\\) with the slope \\(m=\\dfrac{4}{3}\\).<\/p>\r\n\r\n<\/div>\r\n<details><summary class=\"answer\">Show answer<\/summary>\r\n<div id=\"fs-id1169597414728\" data-type=\"solution\"><span id=\"fs-id1169597563595\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_055_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595211024\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595211029\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595340096\" data-type=\"problem\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Graph a line given a point and the slope.<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1169595217425\" class=\"stepwise\" type=\"1\">\r\n \t<li>Plot the given point.<\/li>\r\n \t<li>Use the slope formula \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\) to identify the rise and the run.<\/li>\r\n \t<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\r\n \t<li>Connect the points with a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595339081\" class=\"howto\" data-type=\"note\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597569124\" data-type=\"problem\">\r\n<p id=\"fs-id1169597569127\">Graph the line with <em data-effect=\"italics\">y<\/em>-intercept 2 whose slope is \\(m=-\\dfrac{2}{3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597708562\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1169597423125\">Plot the given point, the <em data-effect=\"italics\">y<\/em>-intercept, \\(\\left(0,2\\right)\\).<\/p>\r\n<span id=\"fs-id1169597739573\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_031_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<table id=\"eip-813\" style=\"height: 64px; width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 171.203px;\">Identify the rise and the run.<\/td>\r\n<td style=\"height: 16px; width: 300.281px;\">\\(m=-\\dfrac{2}{3}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\r\n<td style=\"height: 16px; width: 300.281px;\">\\(\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{-2}{3}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\r\n<td style=\"height: 16px; width: 300.281px;\">\\(\\text{rise}=-2\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\r\n<td style=\"height: 16px; width: 300.281px;\">\\(\\text{run}=3\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169595156199\">Count the rise and the run. Mark the second point.<\/p>\r\n<span id=\"fs-id1169597712731\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled \u201cdown 2\u201d and the line from (0, 0) to (3, 0) is labeled \u201cright 3\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_032_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled \u201cdown 2\u201d and the line from (0, 0) to (3, 0) is labeled \u201cright 3\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169597808478\">Connect the two points with a line.<\/p>\r\n<span id=\"fs-id1169597535041\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169597617902\">You can check your work by finding a third point. Since the slope is \\(m=-\\dfrac{2}{3}\\), it can be written as \\(m=\\dfrac{2}{-3}\\). Go back to \\(\\left(0,2\\right)\\) and count out the rise, 2, and the run, \\(-3\\).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597421959\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597421963\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597753968\" data-type=\"problem\">\r\n<p id=\"fs-id1169597753971\">Graph the line with the <em data-effect=\"italics\">y<\/em>-intercept 4 and slope \\(m=-\\dfrac{5}{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<details><summary class=\"answer\">Show answer<\/summary>\r\n<div id=\"fs-id1169595248307\" data-type=\"solution\"><span id=\"fs-id1169595248310\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_057_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169597489956\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597741331\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595311599\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595152188\" data-type=\"problem\">\r\n<p id=\"fs-id1169595152190\">Graph the line passing through the point \\(\\left(-1,-3\\right)\\) whose slope is \\(m=4.\\)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597525509\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1169595196010\">Plot the given point.<\/p>\r\n<span id=\"fs-id1169595196014\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_034_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.\" data-media-type=\"image\/jpeg\"><\/span>\r\n<table id=\"eip-381\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Identify the rise and the run.<\/td>\r\n<td>\\(m=4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write 4 as a fraction.<\/td>\r\n<td>\\(\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{4}{1}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(\\text{rise}=4,\\text{run}=1\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169597877808\">Count the rise and run and mark the second point.<\/p>\r\n<span id=\"fs-id1169597837045\" data-type=\"media\" data-alt=\"This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_035_img_new.jpg\" alt=\"This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169595222496\">Connect the two points with a line.<\/p>\r\n<span id=\"fs-id1169595222499\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).\" data-media-type=\"image\/jpeg\"><\/span>\r\n<p id=\"fs-id1169597541265\">You can check your work by finding a third point. Since the slope is \\(m=4\\), it can be written as \\(m=\\dfrac{-4}{-1}\\). Go back to \\(\\left(-1,-3\\right)\\) and count out the rise, \\(-4\\), and the run, \\(-1\\).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169597703923\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597703927\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597740280\" data-type=\"problem\">\r\n<p id=\"fs-id1169597740282\">Graph the line with the point \\(\\left(-2,1\\right)\\) and slope \\(m=3\\).<\/p>\r\n\r\n<\/div>\r\n<details><summary class=\"answer\">Show answer<\/summary>\r\n<div id=\"fs-id1169597702694\" data-type=\"solution\"><span id=\"fs-id1169597702698\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_059_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<div class=\"textbox textbox--exercises\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Solve Slope Applications<\/h1>\r\n<p id=\"fs-id1169595255686\">At the beginning of this section, we said there are many applications of slope in the real world. Let\u2019s look at a few now.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595665380\" data-type=\"problem\">\r\n<p id=\"fs-id1169595665382\">The \u2018pitch\u2019 of a building\u2019s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?<\/p>\r\n<span id=\"fs-id1169595305923\" data-type=\"media\" data-alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled &quot;pitch of the roof&quot;. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled &quot;rise equals 9 feet&quot;. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled &quot;run equals 18 feet&quot;.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_037_img_new.jpg\" alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled &quot;pitch of the roof&quot;. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled &quot;rise equals 9 feet&quot;. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled &quot;run equals 18 feet&quot;.\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595150178\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-812\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>\\(m=\\dfrac{\\text{rise}}{\\text{run}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values for rise and run.<\/td>\r\n<td>\\(m=\\dfrac{9}{18}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(m=\\dfrac{1}{2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the roof is \\(\\dfrac{1}{2}\\).<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The roof rises 1 foot for every 2 feet of horizontal run.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595664836\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595664840\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595155539\" data-type=\"problem\">\r\n<p id=\"fs-id1169595155541\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169595665380\">(Example 10)<\/a>, substituting the rise = 14 and run = 24.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595119476\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169595119479\">\\(\\dfrac{7}{12}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169597740030\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169597740034\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595248150\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nHave you ever thought about the sewage pipes going from your house to the street? They must slope down \\(\\dfrac{1}{4}\\) inch per foot in order to drain properly. What is the required slope?\r\n<div id=\"fs-id1169597740546\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597740549\" data-type=\"problem\">\r\n\r\n<span id=\"fs-id1169595155681\" data-type=\"media\" data-alt=\"This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_038_img_new.jpg\" alt=\"This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.\" data-media-type=\"image\/jpeg\"><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169595108140\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<\/div>\r\n<\/div>\r\n<table id=\"eip-496\" style=\"height: 42px; width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 137.406px;\">Use the slope formula.<\/td>\r\n<td style=\"height: 14px; width: 1039.41px;\">\\(\\begin{array}{c}m=\\dfrac{\\text{rise}}{\\text{run}}\\\\ m=\\dfrac{-\\frac{1}{4}\\text{inch}}{\\text{1 foot}}\\\\ m=\\dfrac{-\\frac{1}{4}\\text{inch}}{\\text{12 inches}}\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 137.406px;\">Simplify.<\/td>\r\n<td style=\"height: 14px; width: 1039.41px;\">\\(m=-\\dfrac{1}{48}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 137.406px;\"><\/td>\r\n<td style=\"height: 14px; width: 1039.41px;\">The slope of the pipe is \\(-\\dfrac{1}{48}\\).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe pipe drops 1 inch for every 48 inches of horizontal run.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1169595250216\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1169595250220\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595250222\" data-type=\"problem\">\r\n<p id=\"fs-id1169597836571\">Find the slope of a pipe that slopes down \\(\\dfrac{1}{3}\\) inch per foot.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1169597867411\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\r\n<p id=\"fs-id1169597867641\">\\(-\\dfrac{1}{36}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Key Concepts<\/h1>\r\n<ul>\r\n \t<li><strong>Find the Slope of a Line from its Graph using<\/strong> \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\)\r\n<ol type=\"1\">\r\n \t<li>Locate two points on the line whose coordinates are integers.<\/li>\r\n \t<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\r\n \t<li>Count the rise and the run on the legs of the triangle.<\/li>\r\n \t<li>Take the ratio of rise to run to find the slope.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Graph a Line Given a Point and the Slope<\/strong>\r\n<ol type=\"1\">\r\n \t<li>Plot the given point.<\/li>\r\n \t<li>Use the slope formula \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\) to identify the rise and the run.<\/li>\r\n \t<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\r\n \t<li>Connect the points with a line.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Slope of a Horizontal Line<\/strong>\r\n<ul>\r\n \t<li>The slope of a horizontal line, \\(y=b\\), is 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Slope of a vertical line<\/strong>\r\n<ul>\r\n \t<li>The slope of a vertical line, \\(x=a\\), is undefined<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h1 data-type=\"title\">Glossar<\/h1>\r\n<div class=\"textbox shaded\">\r\n<dl>\r\n \t<dd id=\"fs-id1169595123489\"><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169595147458\">\r\n \t<dt>negative slope<\/dt>\r\n \t<dd id=\"fs-id1169595147463\">A negative slope of a line goes down as you read from left to right.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169595147468\">\r\n \t<dt>positive slope<\/dt>\r\n \t<dd id=\"fs-id1169597687450\">A positive slope of a line goes up as you read from left to right.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169597687455\">\r\n \t<dt>rise<\/dt>\r\n \t<dd id=\"fs-id1169597687460\">The rise of a line is its vertical change.<\/dd>\r\n<\/dl>\r\n<dl>\r\n \t<dt>run<\/dt>\r\n \t<dd id=\"fs-id1169595250173\">The run of a line is its horizontal change.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169595250177\">\r\n \t<dd id=\"fs-id1169595250182\"><strong>slope formula<\/strong><\/dd>\r\n \t<dd>The slope of the line between two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is \\(m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\).<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169597726096\">\r\n \t<dt>slope of a line<\/dt>\r\n \t<dd id=\"fs-id1169597726102\">The slope of a line is \\(m=\\dfrac{\\text{rise}}{\\text{run}}\\). The rise measures the vertical change and the run measures the horizontal change.<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h1 data-type=\"title\">3.4 Exercise Set<\/h1>\r\n<p id=\"fs-id1169595254772\">In the following exercises, find the slope of each line shown.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 481px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 341px;\">\r\n<td style=\"width: 49.3711%; height: 341px;\">\r\n<div data-type=\"problem\">1.<\/div>\r\n<div data-type=\"problem\"><span id=\"fs-id1169595254780\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_215_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\r\n<td style=\"width: 50.6289%; height: 341px;\">\r\n<div id=\"fs-id1169597753202\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div data-type=\"problem\">2.<\/div>\r\n<\/div>\r\n<div class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595663938\" data-type=\"problem\"><span id=\"fs-id1169595663940\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_217_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.3711%; height: 14px;\">\r\n<div id=\"fs-id1169597577035\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div data-type=\"problem\">3.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169597691820\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597691822\" data-type=\"problem\"><span id=\"fs-id1169597691824\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_219_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<td style=\"width: 50.6289%; height: 14px;\">\r\n<div id=\"fs-id1169595223926\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div data-type=\"problem\">4.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595197649\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595197651\" data-type=\"problem\"><span id=\"fs-id1169595197653\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_245_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.3711%; height: 14px;\">\r\n<div id=\"fs-id1169595223446\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div data-type=\"problem\">5.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595303756\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595303758\" data-type=\"problem\"><span id=\"fs-id1169595303760\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_247_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<td style=\"width: 50.6289%; height: 14px;\">\r\n<div id=\"fs-id1169597687073\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597687076\" data-type=\"problem\"><\/div>\r\n<div data-type=\"problem\">6.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169597569721\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597569723\" data-type=\"problem\"><span id=\"fs-id1169595250342\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_249_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.3711%; height: 14px;\">\r\n<div id=\"fs-id1169597740011\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169597705849\" data-type=\"problem\"><\/div>\r\n<div data-type=\"problem\">7.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595270153\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595270156\" data-type=\"problem\"><span id=\"fs-id1169595270158\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_251_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<td style=\"width: 50.6289%; height: 14px;\">\r\n<div id=\"fs-id1169595223382\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div data-type=\"problem\">8.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169595339130\" class=\"material-set-2\" data-type=\"exercise\">\r\n<div id=\"fs-id1169595339132\" data-type=\"problem\"><span id=\"fs-id1169595339134\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_223_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/div>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 id=\"fs-id1169597740086\"><\/h2>\r\n<p id=\"fs-id1169597878413\">In the following exercises, find the slope of each line.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"9\">\r\n \t<li>\\(y=3\\)<\/li>\r\n \t<li>\\(x=4\\)<\/li>\r\n \t<li>\\(y=-2\\)<\/li>\r\n \t<li>\\(x=-5\\)<\/li>\r\n<\/ol>\r\nIn the following exercises, use the slope formula to find the slope of the line between each pair of points.\r\n<ol class=\"twocolumn\" start=\"13\">\r\n \t<li>\\(\\left(1,4\\right),\\left(3,9\\right)\\)<\/li>\r\n \t<li>\\(\\left(0,3\\right),\\left(4,6\\right)\\)<\/li>\r\n \t<li>\\(\\left(2,5\\right),\\left(4,0\\right)\\)<\/li>\r\n \t<li>\\(\\left(-3,3\\right),\\left(4,-5\\right)\\)<\/li>\r\n \t<li>\\(\\left(-1,-2\\right),\\left(2,5\\right)\\)<\/li>\r\n \t<li>\\(\\left(4,-5\\right),\\left(1,-2\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1169595179448\">In the following exercises, graph each line with the given point and slope.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"19\">\r\n \t<li>\\(\\left(1,-2\\right)\\); \\(m=\\dfrac{3}{4}\\)<\/li>\r\n \t<li>\\(\\left(2,5\\right)\\); \\(m=-\\dfrac{1}{3}\\)<\/li>\r\n \t<li>\\(\\left(-3,4\\right)\\); \\(m=-\\dfrac{3}{2}\\)<\/li>\r\n \t<li>\\(\\left(-1,-4\\right)\\); \\(m=\\dfrac{4}{3}\\)<\/li>\r\n \t<li><em data-effect=\"italics\">y<\/em>-intercept 3; \\(m=-\\dfrac{2}{5}\\)<\/li>\r\n \t<li><em data-effect=\"italics\">x<\/em>-intercept \\(-2\\); \\(m=\\dfrac{3}{4}\\)<\/li>\r\n \t<li>\\(\\left(-3,3\\right)\\); \\(m=2\\)<\/li>\r\n \t<li>\\(\\left(1,5\\right)\\); \\(m=-3\\)<\/li>\r\n<\/ol>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p id=\"fs-id1169595363912\">27. An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an \u201c\\(x\\) 12 pitch\u201d meaning \\(\\dfrac{x}{12}\\), where \\(x\\) is the measurement from the roof to the level\u2014the rise. It is also sometimes stated as an \u201c\\(x\\)-in-12 pitch\u201d.<\/p>\r\na) What is the slope of the roof in this picture?\r\n\r\nb) What is the pitch in construction terms?<span data-type=\"newline\">\r\n<\/span> <span id=\"fs-id1169595174151\" data-type=\"media\" data-alt=\"This figure shows one side of a sloped roof of a house. The rise of the roof is labeled \u201c4 inches\u201d and the run of the roof is labeled \u201c12 inches\u201d.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_241_img_new.jpg\" alt=\"This figure shows one side of a sloped roof of a house. The rise of the roof is labeled \u201c4 inches\u201d and the run of the roof is labeled \u201c12 inches\u201d.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"28\">\r\n \t<li data-type=\"title\">A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?<\/li>\r\n \t<li>The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run.\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"a\">\r\n \t<li>How long must the ramp be to accommodate a 24-inch rise to the door?<\/li>\r\n \t<li>Create a model of this ramp.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h2 data-type=\"title\"><span style=\"font-size: 1.2em; font-weight: bold;\">Answers<\/span><\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>\\(\\dfrac{2}{5}\\)<\/li>\r\n \t<li>\\(\\dfrac{5}{4}\\)<\/li>\r\n \t<li>\\(-\\dfrac{1}{3}\\)<\/li>\r\n \t<li>\\(-\\dfrac{3}{4}\\)<\/li>\r\n \t<li>\\(\\dfrac{3}{4}\\)<\/li>\r\n \t<li>\\(-\\dfrac{5}{2}\\)<\/li>\r\n \t<li>\\(-\\dfrac{2}{3}\\)<\/li>\r\n \t<li>\\(\\dfrac{1}{4}\\)<\/li>\r\n \t<li>0<\/li>\r\n \t<li>undefined<\/li>\r\n \t<li>0<\/li>\r\n \t<li>undefined<\/li>\r\n \t<li>\\(\\dfrac{5}{2}\\)<\/li>\r\n \t<li>\\(\\dfrac{3}{4}\\)<\/li>\r\n \t<li>\\(-\\dfrac{5}{2}\\)<\/li>\r\n \t<li>\\(-\\dfrac{8}{7}\\)<\/li>\r\n \t<li>\\(\\dfrac{7}{3}\\)<\/li>\r\n \t<li>\\(-1\\)<\/li>\r\n<\/ol>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 851px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 302px;\">\r\n<td style=\"width: 50%; height: 302px;\">19.\r\n\r\n<span id=\"fs-id1169597837612\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_225_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<td style=\"width: 50%; height: 302px;\">20.\r\n\r\n<span id=\"fs-id1169595287832\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_227_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 302px;\">\r\n<td style=\"width: 50%; height: 302px;\">21.\r\n\r\n<span id=\"fs-id1169597482778\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_229_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<td style=\"width: 50%; height: 302px;\">22.\r\n\r\n<span id=\"fs-id1169597783989\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_231_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">23.\r\n\r\n<span id=\"fs-id1169597741090\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_233_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<td style=\"width: 50%; height: 16px;\">24.\r\n\r\n<span id=\"fs-id1169595340246\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_235_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 50%; height: 16px;\">25.\r\n\r\n<span id=\"fs-id1169597872969\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_237_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<td style=\"width: 50%; height: 16px;\">26.\r\n\r\n<span data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_239_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).\" data-media-type=\"image\/jpeg\"><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol start=\"27\">\r\n \t<li><span class=\"token\">a) <\/span>\\(\\dfrac{1}{3}\\)\u2003b) 4 12 pitch or 4-in-12 pitch<\/li>\r\n \t<li>\\(\\dfrac{3}{50}\\); rise = 3, run = 50<\/li>\r\n \t<li><span class=\"token\">a)<\/span> 288 inches (24 feet)\u2003b) Models will vary.<\/li>\r\n<\/ol>\r\n<h1>Attributions<\/h1>\r\nThis chapter has been adapted from \u201cUnderstand Slope of a Line\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Adaptation Statement for more information.","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to:<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e37065a755a4e447314b671b22cf98b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#114;&#105;&#115;&#101;&#125;&#123;&#114;&#117;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"73\" style=\"vertical-align: -12px;\" \/> to find the slope of a line from its graph<\/li>\n<li>Find the slope of horizontal and vertical lines<\/li>\n<li>Use the slope formula to find the slope of a line between two points<\/li>\n<li>Graph a line given a point and the slope<\/li>\n<li>Solve slope applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"fs-id1169597691001\">When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?<\/p>\n<p id=\"fs-id1169597447394\">In mathematics, the \u2018tilt\u2019 of a line is called the <em data-effect=\"italics\">slope<\/em> of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.<\/p>\n<p id=\"fs-id1169597704220\">In this section, we will explore the concept of slope.<\/p>\n<p id=\"fs-id1169597618132\">The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Slope of a line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169597569399\">The <span data-type=\"term\">slope of a line<\/span> of a line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<p id=\"fs-id1169597775407\">The <span data-type=\"term\">rise<\/span> measures the vertical change and the <span data-type=\"term\">run<\/span> measures the horizontal change between two points on the line.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Positive and negative slopes<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169597508120\">We \u2018read\u2019 a line from left to right just like we read words in English. As you read from left to right, the line\u00a0 is going up; it has <span data-type=\"term\">positive slope<\/span>. The line is going down; it has <span data-type=\"term\">negative slope<\/span>.<\/p>\n<div id=\"fs-id1169595286254\" data-type=\"note\">\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1169597576838\" data-type=\"media\" data-alt=\"The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled \u201cPositive slope\u201d. The line on the right is a diagonal line that drops from left to right. It is labeled \u201cNegative slope\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2020\/08\/CNX_ElemAlg_Figure_04_04_043_img_new.jpg\" alt=\"The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled \u201cPositive slope\u201d. The line on the right is a diagonal line that drops from left to right. It is labeled \u201cNegative slope\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5aa0915be073d2c5b3267688c07240c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/> to Find the Slope of a Line from its Graph<\/h1>\n<p id=\"fs-id1169597708542\">We\u2019ll look at some graphs on the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6f7207ba38a07b82dab0ec08e298c6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: -4px;\" \/>-coordinate plane and see how to find their slopes.<\/p>\n<p id=\"fs-id1169595180855\">To find the slope, we must count out the rise and the run. But where do we start?<\/p>\n<p id=\"fs-id1169597483663\">We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How to Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/> to Find the Slope of a Line from its Graph<\/div>\n<div id=\"fs-id1169595179156\" data-type=\"exercise\">\n<div id=\"fs-id1169597514142\" data-type=\"problem\">\n<p id=\"fs-id1169595237681\">Find the slope of the line shown.<\/p>\n<p><span id=\"fs-id1169597533738\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_013_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169597807891\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1169597422743\" data-type=\"media\" data-alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).\u201d To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061a_img_new.jpg\" alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).\u201d To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597740024\" data-type=\"media\" data-alt=\"The second row says, \u201cStep 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).\u201d In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061b_img_new.jpg\" alt=\"The second row says, \u201cStep 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).\u201d In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597536602\" data-type=\"media\" data-alt=\"The third row then says, \u201cStep 3. Count the rise and the run on the legs of the triangle.\u201d The rise is 4 and the run is 5.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061c_img_new.jpg\" alt=\"The third row then says, \u201cStep 3. Count the rise and the run on the legs of the triangle.\u201d The rise is 4 and the run is 5.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597422236\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.\u201d To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.\u201d To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595313202\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597837782\" data-type=\"exercise\">\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\n<p id=\"fs-id1169597467540\">Find the slope of the line shown.<\/p>\n<p><span id=\"fs-id1169597524912\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_048_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169595663978\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169597825152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4163fe34eaf19a7a1b2db44c29e26eb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Find the slope of a line from its graph using m = rise \/ run.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169597712904\" class=\"stepwise\" type=\"1\">\n<li>Locate two points on the line whose coordinates are integers.<\/li>\n<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\n<li>Count the rise and the run on the legs of the triangle.<\/li>\n<li>Take the ratio of rise to run to find the slope, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595313202\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597837782\" data-type=\"exercise\">\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the slope of the line shown.<\/p>\n<div id=\"fs-id1169595217562\" data-type=\"problem\">\n<p><span id=\"fs-id1169597721249\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_017_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169595310501\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1168465017968\" style=\"width: 100%;\" summary=\"This figure shows step-by-step how to find the slope of the line with points (0, 5) and (3, 3). First, identify the leftmost point, which is (0, 5). Starting at (0, 5), sketch a right triangle to (3, 3). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 9. The y-axis of the plane runs from negative 1 to 7. The points (0, 5) and (3, 3) are plotted. An additional point is plotted at (0, 3). The three points form a right triangle, with the line from (0, 5) to (3, 3) forming the hypotenuse and the lines from (0, 5) to (0, 3) and from (0, 3) to (3, 3) forming the legs. The leg from (0, 5) to (0, 3) is labeled \u201crise\u201d and the leg from (0, 3) to (3, 3) is labeled \u201crun\u201d. The next step is to count the rise, which is negative. The rise is negative 2. The next step is to count the run, which is 3. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals negative 2 thirds.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate two points on the graph whose coordinates are integers.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4a2258b08828b82f5478b79177f57c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Which point is on the left?<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Starting at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>, sketch a right triangle to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4a2258b08828b82f5478b79177f57c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169595186611\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_018a_img_new.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Count the rise\u2014it is negative.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The rise is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Count the run.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The run is 3.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the slope formula.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of the rise and run.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e42fb0686d40a889d8aa18b1744d55b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f73f43cc9463a5d8bd1388705e24bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The slope of the line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7f46d311e63aa198d3e006d236f41ba7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169597467837\">So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> increases by 3 units as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> decreases by 2 units.<\/p>\n<p id=\"fs-id1169597701404\">What if we used the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f03214ae685c15fbb390fe7d48d59247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc9db18ceda8b325515059e9c425b44f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> to find the slope of the line?<\/p>\n<p><span id=\"fs-id1169595297895\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_029_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169595305929\">The rise would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and the run would be 9. Then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-11d02ac7dfc7c36d6df1905f02fe9f83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>, and that simplifies to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f73f43cc9463a5d8bd1388705e24bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>. Remember, it does not matter which points you use\u2014the slope of the line is always the same.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595196353\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597876171\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595339399\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169595180581\">Find the slope of the line shown.<\/p>\n<p><span id=\"fs-id1169595273647\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_050_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169597602160\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169595195547\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf8fd0e597645197b9f8cca5f53db400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169597536164\">In the last two examples, the lines had <em data-effect=\"italics\">y<\/em>-intercepts with integer values, so it was convenient to use the <em data-effect=\"italics\">y<\/em>-intercept as one of the points to find the slope. In the next example, the <em data-effect=\"italics\">y<\/em>-intercept is a fraction. Instead of using that point, we\u2019ll look for two other points whose coordinates are integers. This will make the slope calculations easier.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595185825\" data-type=\"problem\">\n<p id=\"fs-id1169597555840\">Find the slope of the line shown.<\/p>\n<p><span id=\"fs-id1169597555844\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_020_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169597374087\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1168464877438\" style=\"width: 100%;\" summary=\"This figure shows step-by-step how to find the slope of the line with points (2, 3) and (7, 6). First, identify the leftmost point, which is (2, 3). Starting at (2, 3), sketch a right triangle to (7, 6). To the right is the line graphed on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis of the plane runs from 0 to 7. The points (2, 3) and (7, 6) are plotted. An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The leg from (2, 3) to (2, 6) is labeled \u201crise\u201d and the leg from (2, 6) to (7, 6) is labeled \u201crun\u201d. The next step is to count the rise, which is 3. The next step is to count the run, which is 5. Now use the slope formula, m equals rise over run. Substitute the values of the rise and run to get m equals 3 fifths.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate two points on the graph whose coordinates are integers.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Which point is on the left?<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Starting at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>, sketch a right triangle to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169595227774\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_021a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Count the rise.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The rise is 3.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Count the run.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The run is 5.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the slope formula.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of the rise and run.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-810ea15da64891ef91a873a689502d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The slope of the line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-921e5d2a49b3d7a807918f39d7903e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169595287958\">This means that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> increases 5 units as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> increases 3 units.<\/p>\n<p id=\"fs-id1169597838861\">When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.<\/p>\n<p id=\"fs-id1169595312791\">What would happen if we started with the point on the right?<\/p>\n<p id=\"fs-id1169597705426\">Let\u2019s use the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> again, but now we\u2019ll start at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span id=\"fs-id1169595362196\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_022_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-950\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Count the rise.<\/td>\n<td>The rise is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Count the run. It goes from right to left, so it is negative.<\/td>\n<td>The run is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9d4f1414ac1bd4e4c4a67067efcf30da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#125;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The slope of the line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-55756cfaf567459e9a18ae85a43c3a4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#125;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"23\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169595219335\">It does not matter where you start\u2014the slope of the line is always the same.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595120989\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597701127\" data-type=\"exercise\">\n<div id=\"fs-id1169597689410\" data-type=\"problem\">\n<p id=\"fs-id1169597689412\">Find the slope of the line shown.<\/p>\n<p><span data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_052_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169595257080\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169597537337\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f5f811b8571abbd2020c07b55afad264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"9\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Find the Slope of Horizontal and Vertical Lines<\/h1>\n<p id=\"fs-id1169597682080\">Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f02f02ac77847ab445bb35f85dd47c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#72;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#108;&#105;&#110;&#101;&#125;&#32;&#92;&#113;&#117;&#97;&#100;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#121;&#61;&#98;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#86;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#108;&#105;&#110;&#101;&#125;&#92;&#113;&#117;&#97;&#100;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#120;&#61;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#45;&#99;&#111;&#111;&#114;&#100;&#105;&#110;&#97;&#116;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#45;&#99;&#111;&#111;&#114;&#100;&#105;&#110;&#97;&#116;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"442\" style=\"vertical-align: -26px;\" \/><\/p>\n<p id=\"fs-id1169595318009\">So how do we find the slope of the horizontal line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0621c4f761e7864714642fcc62d4c42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/>? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let\u2019s see what happens when we do this.<\/p>\n<p><span id=\"fs-id1169597456359\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_023_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-321\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>What is the rise?<\/td>\n<td>The rise is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>What is the slope?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e8073dfaeecba71044b2b111702534b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#92;&#92;&#32;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#48;&#125;&#123;&#51;&#125;&#92;&#92;&#32;&#109;&#61;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"68\" style=\"vertical-align: -36px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The slope of the horizontal line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0621c4f761e7864714642fcc62d4c42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169597725101\">All horizontal lines have slope 0. When the <em data-effect=\"italics\">y<\/em>-coordinates are the same, the rise is 0.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Slope of a horizontal line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The slope of a horizontal line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfc522d71f8ae0353ab021fa2a90c360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>, is 0.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169597468033\">The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.<\/p>\n<p>Now, we\u2019ll consider a vertical line, the line.<\/p>\n<p><span id=\"fs-id1169595174471\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_024_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-315\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>What is the rise?<\/td>\n<td>The rise is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>What is the slope?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-444b36e9c689058d5bd9f14557f1ac4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#32;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#92;&#92;&#32;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#48;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"71\" width=\"68\" style=\"vertical-align: -31px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169595663765\">But we can\u2019t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> is undefined.<\/p>\n<p id=\"fs-id1169595226611\">The slope of any vertical line is undefined. When the <em data-effect=\"italics\">x<\/em>-coordinates of a line are all the same, the run is 0.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Slope of a vertical line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The slope of a vertical line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2b24e8b3f28f048c85d6ea0f32d59fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\" \/>, is undefined.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595195537\" data-type=\"note\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597714862\" data-type=\"problem\">\n<p id=\"fs-id1169597714865\">Find the slope of each line:<\/p>\n<p id=\"fs-id1169597714866\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1dc438da70f358c2eb1bf64a8b7ea4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>\u2003b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee7bf59b42611ad4534a6d8ca47648e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169597739770\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1168465232289\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1dc438da70f358c2eb1bf64a8b7ea4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><span data-type=\"newline\"><br \/>\n<\/span> This is a vertical line.<span data-type=\"newline\"><br \/>\n<\/span> Its slope is undefined.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span class=\"token\">b)<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee7bf59b42611ad4534a6d8ca47648e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/>\n<\/span> This is a horizontal line.<span data-type=\"newline\"><br \/>\n<\/span> It has slope 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597465640\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597465644\" data-type=\"exercise\">\n<div id=\"fs-id1169597413629\" data-type=\"problem\">\n<p id=\"fs-id1169597413631\">Find the slope of the line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de80293e0b5a1d0b98c28c8b715d136d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169597507381\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169597507383\">undefined<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595176638\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595118607\" data-type=\"exercise\">\n<div id=\"fs-id1169597818064\" data-type=\"solution\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Quick guide to the slopes of lines<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597517831\" data-type=\"note\">\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1169597761828\" data-type=\"media\" data-alt=\"This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_054_img_new.jpg\" alt=\"This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1169597877171\">Remember, we \u2018read\u2019 a line from left to right, just like we read written words in English.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Use the Slope Formula to find the Slope of a Line Between Two Points<\/h1>\n<p id=\"fs-id1169597594129\">Sometimes we\u2019ll need to find the slope of a line between two points when we don\u2019t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we\u2019ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.<\/p>\n<p id=\"fs-id1169597767915\">We have seen that an ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> be used to represent two different points? Mathematicians use subscripts to distinguish the points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6c0866fdb720dc64170f47800cb5877e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#97;&#100;&#32;&#39;&#125;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#120;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#32;&#49;&#44;&#125;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#121;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#32;&#49;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#97;&#100;&#32;&#39;&#125;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#120;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#32;&#50;&#44;&#125;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#32;&#121;&#32;&#92;&#101;&#110;&#115;&#112;&#97;&#99;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#32;&#50;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -16px;\" \/><\/p>\n<p id=\"fs-id1169595174195\">The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?<\/p>\n<p id=\"fs-id1169595219379\">We will use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> to identify the first point and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> to identify the second point.<\/p>\n<p id=\"fs-id1169597700360\">If we had more than two points, we could use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a76652cb71ecde8307c41233858b7e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfa274186e542cb45e885e09262e6015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#52;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/>, and so on.<\/p>\n<p id=\"fs-id1169595147483\">Let\u2019s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span id=\"fs-id1169597494106\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_025_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169597618262\">Since we have two points, we will use subscript notation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bdeb6c9f36981c514bd6d1353e49b85b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#38;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#92;&#32;&#50;&#44;&#32;&#38;&#32;&#51;&#32;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"75\" style=\"vertical-align: -17px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ef7c7240b0bbcc0a71255a146c3f1fd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#32;&#38;&#32;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#55;&#44;&#32;&#38;&#32;&#54;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"75\" style=\"vertical-align: -17px;\" \/>.<\/p>\n<p id=\"fs-id1169597689516\">On the graph, we counted the rise of 3 and the run of 5.<\/p>\n<p id=\"fs-id1169595122971\">Notice that the rise of 3 can be found by subtracting the <em data-effect=\"italics\">y<\/em>-coordinates 6 and 3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b69de98da809c5461d1e0c442528a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#54;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1169595275565\">And the run of 5 can be found by subtracting the <em data-effect=\"italics\">x<\/em>-coordinates 7 and 2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38576c4203754bf70ec4716af3af6697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#61;&#55;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"71\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1169595354853\">We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-810ea15da64891ef91a873a689502d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<p id=\"fs-id1169595344242\">We rewrite the rise and run by putting in the coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00f3910816795998f69d211cd85bb2eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#54;&#45;&#51;&#125;&#123;&#55;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"81\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<p id=\"fs-id1169595119525\">But 6 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e09b16903e98422ba9e190869e1901c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/>, the <em data-effect=\"italics\">y<\/em>-coordinate of the second point and 3 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>, the <em data-effect=\"italics\">y<\/em>-coordinate of the first point.<\/p>\n<p id=\"fs-id1169597723209\">So we can rewrite the slope using subscript notation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40f9d4957ef8442a96898fa467208c5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#55;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"96\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1169595311347\">Also, 7 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-045b76d969dd32d5d1a7688d55a5d858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/>, the <em data-effect=\"italics\">x<\/em>-coordinate of the second point and 2 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1871183b5210fb48e0725365962abeb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/>, the <em data-effect=\"italics\">x<\/em>-coordinate of the first point.<\/p>\n<p id=\"fs-id1169595213803\">So, again, we rewrite the slope using subscript notation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/><\/p>\n<p id=\"fs-id1169595308709\">We\u2019ve shown that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/> is really another version of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>. We can use this formula to find the slope of a line when we have two points on the line.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Slope formula<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169597507992\">The slope of the line between two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/><\/p>\n<p id=\"fs-id1169595287765\">This is the <span data-type=\"term\">slope formula<\/span>.<\/p>\n<p id=\"fs-id1169595255399\">The slope is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3ba6a6a763f3db54daed62d663645f98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#92;&#32;&#32;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#112;&#111;&#105;&#110;&#116;&#32;&#109;&#105;&#110;&#117;&#115;&#32;&#121;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#112;&#111;&#105;&#110;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#118;&#101;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#112;&#111;&#105;&#110;&#116;&#32;&#109;&#105;&#110;&#117;&#115;&#32;&#120;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#112;&#111;&#105;&#110;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"368\" style=\"vertical-align: -37px;\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169597538694\" data-type=\"note\">\n<div id=\"fs-id1169595255402\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595125760\" data-type=\"problem\">\n<p id=\"fs-id1169597721158\">Use the <span class=\"no-emphasis\" data-type=\"term\">slope formula<\/span> to find the slope of the line between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ce830c2dfa3b70e2906cf4d1b7248973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b385c1ebb27da6a9f60a8f21a49f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169597537093\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-758\" style=\"height: 160px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 303.438px;\">We&#8217;ll call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ce830c2dfa3b70e2906cf4d1b7248973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> point #1 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b385c1ebb27da6a9f60a8f21a49f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> point #2.<\/td>\n<td style=\"height: 32px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-28a0209b00bad828df0a1a52acadabf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#38;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#92;&#32;&#49;&#44;&#32;&#38;&#32;&#50;&#32;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"75\" style=\"vertical-align: -17px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2d4a0656b7ed7df15eae8e8cc502916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#32;&#38;&#32;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#52;&#44;&#32;&#38;&#32;&#53;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"75\" style=\"vertical-align: -17px;\" \/>.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 303.438px;\">Use the slope formula.<\/td>\n<td style=\"height: 16px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/>.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 303.438px;\">Substitute the values.<\/td>\n<td style=\"height: 16px; width: 546.781px;\"><\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 303.438px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of the second point minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of the first point<\/td>\n<td style=\"height: 32px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5b5fcf1b60a20db0a6977029153aff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#45;&#50;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"99\" style=\"vertical-align: -15px;\" \/>.<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 303.438px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> of the second point minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> of the first point<\/td>\n<td style=\"height: 32px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ec8181112c08e64a04834a2136d65f0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#45;&#50;&#125;&#123;&#52;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"81\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 303.438px;\">Simplify the numerator and the denominator.<\/td>\n<td style=\"height: 16px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4f0404c4a68ff0a839a5c706e9b6e239_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 303.438px;\">Simplify.<\/td>\n<td style=\"height: 16px; width: 546.781px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d94dded0dc9c883f82d566d62e2d4b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169595104074\">Let\u2019s confirm this by counting out the slope on a graph using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<p><span id=\"fs-id1169597690980\" data-type=\"media\" data-alt=\"The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled \u201crise\u201d and the leg from (1, 5) to (4, 5) is labeled \u201crun\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_026_img_new.jpg\" alt=\"The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled \u201crise\u201d and the leg from (1, 5) to (4, 5) is labeled \u201crun\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169595176270\">It doesn\u2019t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595362728\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597805559\" data-type=\"exercise\">\n<div id=\"fs-id1169597805561\" data-type=\"problem\">\n<p id=\"fs-id1169597805563\">Use the slope formula to find the slope of the line through the points: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-648acaf5866f6160ec660bee40426678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f1a235ff31c373d5cda1965ad172871_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169595123179\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169595123181\">1<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595309361\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597836954\" data-type=\"exercise\">\n<div id=\"fs-id1169597603850\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597740105\" data-type=\"problem\">\n<p id=\"fs-id1169597740107\">Use the slope formula to find the slope of the line through the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e6b25c6279fe53efe7c78796a156eeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-587b1474fbf6d7b7f8be76f9d4c02c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169597807761\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-471\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We&#8217;ll call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e6b25c6279fe53efe7c78796a156eeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> point #1 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-587b1474fbf6d7b7f8be76f9d4c02c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> point #2.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3beee36edd774556d1917fcc807fdfe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#38;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#92;&#32;&#45;&#50;&#44;&#32;&#38;&#32;&#45;&#51;&#32;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"86\" style=\"vertical-align: -17px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2b11abac1ff39a6eb0ba878a58dd8ee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#32;&#38;&#32;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#45;&#55;&#44;&#32;&#38;&#32;&#52;&#92;&#101;&#110;&#100;&#123;&#112;&#109;&#97;&#116;&#114;&#105;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"80\" style=\"vertical-align: -17px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of the second point minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of the first point<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d01f5a676870f925aa8c98262a59ba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"108\" style=\"vertical-align: -15px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> of the second point minus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> of the first point<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2653326eec7033e3d0c5b45f79f95b01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"122\" style=\"vertical-align: -17px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c5083f1a446715fffacc6ca187972465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#45;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"74\" width=\"64\" style=\"vertical-align: -32px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169597814147\">Let\u2019s verify this slope on the graph shown.<\/p>\n<p><span id=\"fs-id1169595106906\" data-type=\"media\" data-alt=\"The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled \u201crise\u201d and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled \u201crun\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_027_img_new.jpg\" alt=\"The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled \u201crise\u201d and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled \u201crun\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div id=\"fs-id1169597824078\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-31cbf95ba1d99da2192a766cd2d456e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#55;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"91\" style=\"vertical-align: -49px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595228085\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595228089\" data-type=\"exercise\">\n<div id=\"fs-id1169597508758\" data-type=\"problem\">\n<p id=\"fs-id1169597508760\">Use the slope formula to find the slope of the line through the points: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-862a9525ac8db19009bf877fff4597b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9ed3f84e4d9fd37e7c85bf5177357fe0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169597821110\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169597821112\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Graph a Line Given a Point and the Slope<\/h1>\n<p id=\"fs-id1169597413558\">Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.<\/p>\n<p id=\"fs-id1169597421283\">One other method we can use to graph lines is called the <span data-type=\"term\">point\u2013slope method<\/span>. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How To Graph a Line Given a Point and The Slope<\/div>\n<div id=\"fs-id1169595256096\" data-type=\"exercise\">\n<div id=\"fs-id1169597466646\" data-type=\"problem\">\n<p id=\"fs-id1169597773540\">Graph the line passing through the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c3ac38dbb39343c28a60a287dfb114b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> whose slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c779c458c4158772b612729d0ca341b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169595176366\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1169597715877\" data-type=\"media\" data-alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Plot the given point. Plot (1, negative 1).\u201d To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062a_img_new.jpg\" alt=\"This table has three columns and four rows. The first row says, \u201cStep 1. Plot the given point. Plot (1, negative 1).\u201d To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597430242\" data-type=\"media\" data-alt=\"The second row says, \u201cStep 2. Use the slope formula m equals rise divided by run to identify the rise and the run.\u201d The rise and run are 3 and 4, so m equals 3 divided by 4.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062b_img_new.jpg\" alt=\"The second row says, \u201cStep 2. Use the slope formula m equals rise divided by run to identify the rise and the run.\u201d The rise and run are 3 and 4, so m equals 3 divided by 4.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597521254\" data-type=\"media\" data-alt=\"The third row says \u201cStep 3. Starting at the given point, count out the rise and run to mark the second point.\u201d We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062c_img_new.jpg\" alt=\"The third row says \u201cStep 3. Starting at the given point, count out the rise and run to mark the second point.\u201d We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1169597331408\" data-type=\"media\" data-alt=\"The fourth row says \u201cStep 4. Connect the points with a line.\u201d On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062d_img_new.jpg\" alt=\"The fourth row says \u201cStep 4. Connect the points with a line.\u201d On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595211024\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595211029\" data-type=\"exercise\">\n<div id=\"fs-id1169595340096\" data-type=\"problem\">\n<p id=\"fs-id1169595340098\">Graph the line passing through the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8fbb5d034d8bc7481119846ba5facddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-70fe2977b04eb74081a54e4989b4e637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<\/div>\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<div id=\"fs-id1169597414728\" data-type=\"solution\"><span id=\"fs-id1169597563595\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_055_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595211024\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595211029\" data-type=\"exercise\">\n<div id=\"fs-id1169595340096\" data-type=\"problem\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Graph a line given a point and the slope.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169595217425\" class=\"stepwise\" type=\"1\">\n<li>Plot the given point.<\/li>\n<li>Use the slope formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/> to identify the rise and the run.<\/li>\n<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\n<li>Connect the points with a line.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169595339081\" class=\"howto\" data-type=\"note\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597569124\" data-type=\"problem\">\n<p id=\"fs-id1169597569127\">Graph the line with <em data-effect=\"italics\">y<\/em>-intercept 2 whose slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f73f43cc9463a5d8bd1388705e24bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169597708562\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169597423125\">Plot the given point, the <em data-effect=\"italics\">y<\/em>-intercept, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-398828550549fdd4b2191f8f7cde7bd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span id=\"fs-id1169597739573\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_031_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-813\" style=\"height: 64px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 171.203px;\">Identify the rise and the run.<\/td>\n<td style=\"height: 16px; width: 300.281px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f73f43cc9463a5d8bd1388705e24bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\n<td style=\"height: 16px; width: 300.281px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-af27ce886b5ad397329f57254b2b42b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"77\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\n<td style=\"height: 16px; width: 300.281px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2ce7a092feeee613e1ebe26a9eda73df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 171.203px;\"><\/td>\n<td style=\"height: 16px; width: 300.281px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d8d8a1bb415bd58c988e4436f7f943e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169595156199\">Count the rise and the run. Mark the second point.<\/p>\n<p><span id=\"fs-id1169597712731\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled \u201cdown 2\u201d and the line from (0, 0) to (3, 0) is labeled \u201cright 3\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_032_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled \u201cdown 2\u201d and the line from (0, 0) to (3, 0) is labeled \u201cright 3\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169597808478\">Connect the two points with a line.<\/p>\n<p><span id=\"fs-id1169597535041\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_061_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169597617902\">You can check your work by finding a third point. Since the slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f73f43cc9463a5d8bd1388705e24bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>, it can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8ca760e5f528ac170c6987e853527ff8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>. Go back to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-398828550549fdd4b2191f8f7cde7bd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and count out the rise, 2, and the run, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597421959\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597421963\" data-type=\"exercise\">\n<div id=\"fs-id1169597753968\" data-type=\"problem\">\n<p id=\"fs-id1169597753971\">Graph the line with the <em data-effect=\"italics\">y<\/em>-intercept 4 and slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1e7c057d4d9b1c7b38a4eeec750d742c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"70\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<div id=\"fs-id1169595248307\" data-type=\"solution\"><span id=\"fs-id1169595248310\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_057_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169597489956\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597741331\" data-type=\"exercise\">\n<div id=\"fs-id1169595311599\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595152188\" data-type=\"problem\">\n<p id=\"fs-id1169595152190\">Graph the line passing through the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-659f5d2dee5bcea4b83fdb4d330c9b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> whose slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6d260f9beb6a6f99f777705e632f246_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169597525509\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169595196010\">Plot the given point.<\/p>\n<p><span id=\"fs-id1169595196014\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_034_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-381\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Identify the rise and the run.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write 4 as a fraction.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-157e880658c5934aa447312fa642bf27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"63\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24f737d9d3a4fd183271bb5b1f5a1471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#61;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169597877808\">Count the rise and run and mark the second point.<\/p>\n<p><span id=\"fs-id1169597837045\" data-type=\"media\" data-alt=\"This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_035_img_new.jpg\" alt=\"This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169595222496\">Connect the two points with a line.<\/p>\n<p><span id=\"fs-id1169595222499\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_062_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169597541265\">You can check your work by finding a third point. Since the slope is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/>, it can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e1d00ad7bea5db3d71422a764eb127b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#125;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/>. Go back to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-659f5d2dee5bcea4b83fdb4d330c9b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and count out the rise, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/>, and the run, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169597703923\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597703927\" data-type=\"exercise\">\n<div id=\"fs-id1169597740280\" data-type=\"problem\">\n<p id=\"fs-id1169597740282\">Graph the line with the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-260107cba86a7b21e919180b1130050e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<div id=\"fs-id1169597702694\" data-type=\"solution\"><span id=\"fs-id1169597702698\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_059_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"solution\">\n<div class=\"textbox textbox--exercises\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Solve Slope Applications<\/h1>\n<p id=\"fs-id1169595255686\">At the beginning of this section, we said there are many applications of slope in the real world. Let\u2019s look at a few now.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595665380\" data-type=\"problem\">\n<p id=\"fs-id1169595665382\">The \u2018pitch\u2019 of a building\u2019s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?<\/p>\n<p><span id=\"fs-id1169595305923\" data-type=\"media\" data-alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled &quot;pitch of the roof&quot;. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled &quot;rise equals 9 feet&quot;. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled &quot;run equals 18 feet&quot;.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_037_img_new.jpg\" alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled &quot;pitch of the roof&quot;. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled &quot;rise equals 9 feet&quot;. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled &quot;run equals 18 feet&quot;.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169595150178\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-812\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Use the slope formula.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Substitute the values for rise and run.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-23eadf6eeb7052ebfd5ac236f4309f9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"59\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5ef41f038f64d1359e49d5ea433122d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The slope of the roof is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5aa3fe0ad89cf65f34b5c880c078356e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The roof rises 1 foot for every 2 feet of horizontal run.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595664836\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595664840\" data-type=\"exercise\">\n<div id=\"fs-id1169595155539\" data-type=\"problem\">\n<p id=\"fs-id1169595155541\">Use <a class=\"autogenerated-content\" href=\"#fs-id1169595665380\">(Example 10)<\/a>, substituting the rise = 14 and run = 24.<\/p>\n<\/div>\n<div id=\"fs-id1169595119476\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169595119479\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea9aa58714d2567cc4c8ae47c88c21e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"18\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169597740030\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169597740034\" data-type=\"exercise\">\n<div id=\"fs-id1169595248150\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Have you ever thought about the sewage pipes going from your house to the street? They must slope down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7fc58100d71714be3c30d3c3b33007fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/> inch per foot in order to drain properly. What is the required slope?<\/p>\n<div id=\"fs-id1169597740546\" data-type=\"exercise\">\n<div id=\"fs-id1169597740549\" data-type=\"problem\">\n<p><span id=\"fs-id1169595155681\" data-type=\"media\" data-alt=\"This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_038_img_new.jpg\" alt=\"This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1169595108140\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<\/div>\n<\/div>\n<table id=\"eip-496\" style=\"height: 42px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 137.406px;\">Use the slope formula.<\/td>\n<td style=\"height: 14px; width: 1039.41px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6a57170ca539d8ccf250f927a6f50d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;&#92;&#92;&#32;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#99;&#104;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#102;&#111;&#111;&#116;&#125;&#125;&#92;&#92;&#32;&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#99;&#104;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#105;&#110;&#99;&#104;&#101;&#115;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"112\" style=\"vertical-align: -54px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 137.406px;\">Simplify.<\/td>\n<td style=\"height: 14px; width: 1039.41px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b2cdd3783ebe32edbf328baf6f8393f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"73\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 137.406px;\"><\/td>\n<td style=\"height: 14px; width: 1039.41px;\">The slope of the pipe is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-adeb025962d555b723420431f027e478_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"33\" style=\"vertical-align: -12px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The pipe drops 1 inch for every 48 inches of horizontal run.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169595250216\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169595250220\" data-type=\"exercise\">\n<div id=\"fs-id1169595250222\" data-type=\"problem\">\n<p id=\"fs-id1169597836571\">Find the slope of a pipe that slopes down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e1435fbb34dc827f8ae4d908bf2364d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/> inch per foot.<\/p>\n<\/div>\n<div id=\"fs-id1169597867411\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1169597867641\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8efb97228581bdbbc181b27e3da6107b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"33\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul>\n<li><strong>Find the Slope of a Line from its Graph using<\/strong> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>\n<ol type=\"1\">\n<li>Locate two points on the line whose coordinates are integers.<\/li>\n<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\n<li>Count the rise and the run on the legs of the triangle.<\/li>\n<li>Take the ratio of rise to run to find the slope.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Graph a Line Given a Point and the Slope<\/strong>\n<ol type=\"1\">\n<li>Plot the given point.<\/li>\n<li>Use the slope formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/> to identify the rise and the run.<\/li>\n<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\n<li>Connect the points with a line.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Slope of a Horizontal Line<\/strong>\n<ul>\n<li>The slope of a horizontal line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfc522d71f8ae0353ab021fa2a90c360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>, is 0.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Slope of a vertical line<\/strong>\n<ul>\n<li>The slope of a vertical line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2b24e8b3f28f048c85d6ea0f32d59fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\" \/>, is undefined<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1 data-type=\"title\">Glossar<\/h1>\n<div class=\"textbox shaded\">\n<dl>\n<dd id=\"fs-id1169595123489\"><\/dd>\n<\/dl>\n<dl id=\"fs-id1169595147458\">\n<dt>negative slope<\/dt>\n<dd id=\"fs-id1169595147463\">A negative slope of a line goes down as you read from left to right.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169595147468\">\n<dt>positive slope<\/dt>\n<dd id=\"fs-id1169597687450\">A positive slope of a line goes up as you read from left to right.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169597687455\">\n<dt>rise<\/dt>\n<dd id=\"fs-id1169597687460\">The rise of a line is its vertical change.<\/dd>\n<\/dl>\n<dl>\n<dt>run<\/dt>\n<dd id=\"fs-id1169595250173\">The run of a line is its horizontal change.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169595250177\">\n<dd id=\"fs-id1169595250182\"><strong>slope formula<\/strong><\/dd>\n<dd>The slope of the line between two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3f73755cfea11ad6340a9ab624a8629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"99\" style=\"vertical-align: -15px;\" \/>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169597726096\">\n<dt>slope of a line<\/dt>\n<dd id=\"fs-id1169597726102\">The slope of a line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7bf3b27f8bf32d800686f5f7ddef6962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#105;&#115;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#117;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"68\" style=\"vertical-align: -12px;\" \/>. The rise measures the vertical change and the run measures the horizontal change.<\/dd>\n<\/dl>\n<\/div>\n<h1 data-type=\"title\">3.4 Exercise Set<\/h1>\n<p id=\"fs-id1169595254772\">In the following exercises, find the slope of each line shown.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 481px;\">\n<tbody>\n<tr style=\"height: 341px;\">\n<td style=\"width: 49.3711%; height: 341px;\">\n<div data-type=\"problem\">1.<\/div>\n<div data-type=\"problem\"><span id=\"fs-id1169595254780\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_215_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 50.6289%; height: 341px;\">\n<div id=\"fs-id1169597753202\" class=\"material-set-2\" data-type=\"exercise\">\n<div data-type=\"problem\">2.<\/div>\n<\/div>\n<div class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169595663938\" data-type=\"problem\"><span id=\"fs-id1169595663940\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_217_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.3711%; height: 14px;\">\n<div id=\"fs-id1169597577035\" class=\"material-set-2\" data-type=\"exercise\">\n<div data-type=\"problem\">3.<\/div>\n<\/div>\n<div id=\"fs-id1169597691820\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169597691822\" data-type=\"problem\"><span id=\"fs-id1169597691824\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_219_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<td style=\"width: 50.6289%; height: 14px;\">\n<div id=\"fs-id1169595223926\" class=\"material-set-2\" data-type=\"exercise\">\n<div data-type=\"problem\">4.<\/div>\n<\/div>\n<div id=\"fs-id1169595197649\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169595197651\" data-type=\"problem\"><span id=\"fs-id1169595197653\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_245_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.3711%; height: 14px;\">\n<div id=\"fs-id1169595223446\" class=\"material-set-2\" data-type=\"exercise\">\n<div data-type=\"problem\">5.<\/div>\n<\/div>\n<div id=\"fs-id1169595303756\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169595303758\" data-type=\"problem\"><span id=\"fs-id1169595303760\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_247_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<td style=\"width: 50.6289%; height: 14px;\">\n<div id=\"fs-id1169597687073\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169597687076\" data-type=\"problem\"><\/div>\n<div data-type=\"problem\">6.<\/div>\n<\/div>\n<div id=\"fs-id1169597569721\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169597569723\" data-type=\"problem\"><span id=\"fs-id1169595250342\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_249_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.3711%; height: 14px;\">\n<div id=\"fs-id1169597740011\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169597705849\" data-type=\"problem\"><\/div>\n<div data-type=\"problem\">7.<\/div>\n<\/div>\n<div id=\"fs-id1169595270153\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169595270156\" data-type=\"problem\"><span id=\"fs-id1169595270158\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_251_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<td style=\"width: 50.6289%; height: 14px;\">\n<div id=\"fs-id1169595223382\" class=\"material-set-2\" data-type=\"exercise\">\n<div data-type=\"problem\">8.<\/div>\n<\/div>\n<div id=\"fs-id1169595339130\" class=\"material-set-2\" data-type=\"exercise\">\n<div id=\"fs-id1169595339132\" data-type=\"problem\"><span id=\"fs-id1169595339134\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_223_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1169597740086\"><\/h2>\n<p id=\"fs-id1169597878413\">In the following exercises, find the slope of each line.<\/p>\n<ol class=\"twocolumn\" start=\"9\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8e36d35d8563f5053efd9935e88634f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f986dbfac9d3f29a18cba91e9efa9d2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3e213d8d687e32831c24e16c432b60e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<p>In the following exercises, use the slope formula to find the slope of the line between each pair of points.<\/p>\n<ol class=\"twocolumn\" start=\"13\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-91982c390a8b080583493b737149b4f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1e40766112da9e99440f5b47880e7ae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4fa20f2c57208b106135b7afa640cc63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ded9603cbeb6bf787785e6d7ab7587fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-821b62e43de7af525ced778244f0dc46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b82016ecd1b5d853b61f935d625801ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1169595179448\">In the following exercises, graph each line with the given point and slope.<\/p>\n<ol class=\"twocolumn\" start=\"19\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a685e6a8556e8135bece1243dc70fe1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c779c458c4158772b612729d0ca341b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2da1eb750fc283f55cb9396d5536b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-534178d0d3fffb2edec17b7c9dc333df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-862a9525ac8db19009bf877fff4597b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c82d1186e47d35ce2d9ed2bc931c1c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8235f4c83db17d3a3454713a44752b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-70fe2977b04eb74081a54e4989b4e637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><em data-effect=\"italics\">y<\/em>-intercept 3; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8cd6975e33735ed1061f5783198c10e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"64\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><em data-effect=\"italics\">x<\/em>-intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c779c458c4158772b612729d0ca341b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"50\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03125d47f1e456b2a5c3479daa28f59c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1ee3bb14bbe97a1114d697f8b45a9f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc88778f80c6fa1eb186cfd3741de73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-53a904699623d35001768f429bb03314_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"62\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1169595363912\">27. An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an \u201c<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> 12 pitch\u201d meaning <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-838340f83bd747e90e6edd068fab74ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"18\" style=\"vertical-align: -12px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is the measurement from the roof to the level\u2014the rise. It is also sometimes stated as an \u201c<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-in-12 pitch\u201d.<\/p>\n<p>a) What is the slope of the roof in this picture?<\/p>\n<p>b) What is the pitch in construction terms?<span data-type=\"newline\"><br \/>\n<\/span> <span id=\"fs-id1169595174151\" data-type=\"media\" data-alt=\"This figure shows one side of a sloped roof of a house. The rise of the roof is labeled \u201c4 inches\u201d and the run of the roof is labeled \u201c12 inches\u201d.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_241_img_new.jpg\" alt=\"This figure shows one side of a sloped roof of a house. The rise of the roof is labeled \u201c4 inches\u201d and the run of the roof is labeled \u201c12 inches\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"28\">\n<li data-type=\"title\">A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?<\/li>\n<li>The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run.\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>How long must the ramp be to accommodate a 24-inch rise to the door?<\/li>\n<li>Create a model of this ramp.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h2 data-type=\"title\"><span style=\"font-size: 1.2em; font-weight: bold;\">Answers<\/span><\/h2>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4163fe34eaf19a7a1b2db44c29e26eb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f5f811b8571abbd2020c07b55afad264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"9\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-88835b3a5f91c81e9d1de8b6bad1633f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fcb3955713d0734f495f471ac217fdaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c5ebd9d3766fcdac5860e5f2039b3782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"9\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d443dea57c9f4a55b5718e1ff410d26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7f46d311e63aa198d3e006d236f41ba7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7fc58100d71714be3c30d3c3b33007fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/><\/li>\n<li>0<\/li>\n<li>undefined<\/li>\n<li>0<\/li>\n<li>undefined<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4edbebb5cbb9d375a2aaccf92c5991ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"9\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c5ebd9d3766fcdac5860e5f2039b3782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"9\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d443dea57c9f4a55b5718e1ff410d26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d1a24697af0a73ca0e2805e697905a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"24\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-45c9acdf434791b24a91026a516113e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"9\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%; height: 851px;\">\n<tbody>\n<tr style=\"height: 302px;\">\n<td style=\"width: 50%; height: 302px;\">19.<\/p>\n<p><span id=\"fs-id1169597837612\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_225_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 302px;\">20.<\/p>\n<p><span id=\"fs-id1169595287832\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_227_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 302px;\">\n<td style=\"width: 50%; height: 302px;\">21.<\/p>\n<p><span id=\"fs-id1169597482778\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_229_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 302px;\">22.<\/p>\n<p><span id=\"fs-id1169597783989\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_231_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">23.<\/p>\n<p><span id=\"fs-id1169597741090\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_233_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 16px;\">24.<\/p>\n<p><span id=\"fs-id1169595340246\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_235_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">25.<\/p>\n<p><span id=\"fs-id1169597872969\" data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_237_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%; height: 16px;\">26.<\/p>\n<p><span data-type=\"media\" data-alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_04_04_239_img_new.jpg\" alt=\"The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"27\">\n<li><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e1435fbb34dc827f8ae4d908bf2364d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"9\" style=\"vertical-align: -12px;\" \/>\u2003b) 4 12 pitch or 4-in-12 pitch<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b76f77cc963613b675e14511b29efdd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"18\" style=\"vertical-align: -12px;\" \/>; rise = 3, run = 50<\/li>\n<li><span class=\"token\">a)<\/span> 288 inches (24 feet)\u2003b) Models will vary.<\/li>\n<\/ol>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cUnderstand Slope of a Line\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Adaptation Statement for more information.<\/p>\n","protected":false},"author":125,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1017","chapter","type-chapter","status-publish","hentry"],"part":777,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1017","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1017\/revisions"}],"predecessor-version":[{"id":2174,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1017\/revisions\/2174"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/777"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1017\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1017"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1017"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1017"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}