{"id":1136,"date":"2019-07-29T23:24:31","date_gmt":"2019-07-29T23:24:31","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-graphing\/"},"modified":"2021-08-31T21:20:28","modified_gmt":"2021-08-31T21:20:28","slug":"solve-systems-of-equations-by-graphing","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-graphing\/","title":{"raw":"4.1  Solve Systems of Equations by Graphing","rendered":"4.1  Solve Systems of Equations by Graphing"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section it is expected that you will be able to:\n<ul>\n \t<li>Determine whether an ordered pair is a solution of a system of equations<\/li>\n \t<li>Solve a system of linear equations by graphing<\/li>\n \t<li>Determine the number of solutions of linear system<\/li>\n \t<li>Solve applications of systems of equations by graphing<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Determine Whether an Ordered Pair is a Solution of a System of Equations<\/h1>\n<div id=\"fs-id1168341923821\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345398220\">\u00a0We learned before how to solve linear equations with one variable. Now we will work with <span data-type=\"term\">systems of linear equations<\/span>, two or more linear equations grouped together, witch is known as a system of linear equations.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">System of Linear Equations<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nWhen two or more linear equations are grouped together, they form a system of linear equations.\n\n<\/div>\n<\/div>\n<p id=\"fs-id1168345269961\">We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.<\/p>\n<p id=\"fs-id1168341857638\">An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.<\/p>\n\n<div id=\"fs-id1168345242279\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345435392\">A linear equation in two variables, like 2<em data-effect=\"italics\">x<\/em> + <em data-effect=\"italics\">y<\/em> = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.<\/p>\n<p id=\"fs-id1168345385750\">To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that make both equations true. These are called the <span class=\"no-emphasis\" data-type=\"term\">solutions to a system of equations<\/span>.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Solutions of a System of Equations<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\n<strong data-effect=\"bold\">Solutions of a system of equations<\/strong> are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>).\n\n<\/div>\n<\/div>\n<p id=\"fs-id1168345430694\">To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.<\/p>\n<p id=\"fs-id1168345561194\">Let\u2019s consider the system below:<\/p>\n\n<div id=\"fs-id1168345250439\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}3x-y=7\\hfill \\\\ x-2y=4\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345452720\">Is the ordered pair \\(\\left(2,-1\\right)\\) a solution?<\/p>\n<span id=\"fs-id1168345451927\" data-type=\"media\" data-alt=\"This figure begins with a sentence, \u201cWe substitute x =2 and y = -1 into both equations.\u201d The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_01_001_img_new.jpg\" alt=\"This figure begins with a sentence, \u201cWe substitute x =2 and y = -1 into both equations.\u201d The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345741629\">The ordered pair (2, \u22121) made both equations true. Therefore (2, \u22121) is a solution to this system.<\/p>\n<p id=\"fs-id1168345273771\">Let\u2019s try another ordered pair. Is the ordered pair (3, 2) a solution?<\/p>\n<span id=\"fs-id1168345745133\" data-type=\"media\" data-alt=\"This figure begins with the sentence, \u201cWe substitute x equals 3 and y equals 2 into both equations.\u201d The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_002_img_new.jpg\" alt=\"This figure begins with the sentence, \u201cWe substitute x equals 3 and y equals 2 into both equations.\u201d The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345284173\">The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to <strong data-effect=\"bold\">both<\/strong> equations, it is not a solution to this system.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345276323\" data-type=\"problem\">\n<p id=\"fs-id1168345375020\">Determine whether the ordered pair is a solution to the system: \\(\\left\\{\\begin{array}{c}x-y=-1\\hfill \\\\ 2x-y=-5\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1168345296934\"><span class=\"token\">a) <\/span>\\(\\left(-2,-1\\right)\\) b) \\(\\left(-4,-3\\right)\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345356556\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n\n<span data-type=\"newline\">a)\n<\/span><span id=\"fs-id1168345219786\" data-type=\"media\" data-alt=\"This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, \u201cWe substitute x = negative 2 and y = 1 into both equations,\u201d follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, \u201c(negative 2, negative 1) does not make both equations true.\u201d Under the second equation is the sentence, \u201c(negative 2, negative 1) is not a solution.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_003_img_new.jpg\" alt=\"This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, \u201cWe substitute x = negative 2 and y = 1 into both equations,\u201d follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, \u201c(negative 2, negative 1) does not make both equations true.\u201d Under the second equation is the sentence, \u201c(negative 2, negative 1) is not a solution.\u201d\" data-media-type=\"image\/jpeg\"><\/span><span data-type=\"newline\">\n<\/span>(\u22122, \u22121) does not make both equations true. (\u22122, \u22121) is not a solution.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345509987\" data-type=\"media\" data-alt=\"This figure begins with the sentence, \u201cWe substitute x = -4 and y = -3 into both equations.\u201d The first equation listed shows x \u2013 y = -1. Then -4 - (-3) = -1. Then -1 = -1. The second equation listed shows 2x \u2013 y = -5. Then 2 times (-4) \u2013 (-3) = -5. Then -5 = -5. Under the first equation is the sentence, \u201c(-4, -3) does make both equations true.\u201d Under the second equation is the sentence, \u201c(-4, -3) is a solution.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_004_img_new.jpg\" alt=\"This figure begins with the sentence, \u201cWe substitute x = -4 and y = -3 into both equations.\u201d The first equation listed shows x \u2013 y = -1. Then -4 - (-3) = -1. Then -1 = -1. The second equation listed shows 2x \u2013 y = -5. Then 2 times (-4) \u2013 (-3) = -5. Then -5 = -5. Under the first equation is the sentence, \u201c(-4, -3) does make both equations true.\u201d Under the second equation is the sentence, \u201c(-4, -3) is a solution.\u201d\" data-media-type=\"image\/jpeg\"><\/span><span data-type=\"newline\">\n<\/span>(\u22124, \u22123) does not make both equations true. (\u22124, \u22123) is a solution.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345425152\" data-type=\"problem\">\n<p id=\"fs-id1168345436547\">Determine whether the ordered pair is a solution to the system: \\(\\left\\{\\begin{array}{c}3x+y=0\\hfill \\\\ x+2y=-5\\hfill \\end{array}.\\)<\/p>\n<p id=\"fs-id1168341916968\"><span class=\"token\">a) <\/span>\\(\\left(1,-3\\right)\\) b) \\(\\left(0,0\\right)\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345441011\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345261680\"><span class=\"token\">a)<\/span> yes b) no<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<h1>Equations by Graphing<\/h1>\n<\/div>\n<div id=\"fs-id1168345440846\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345425485\">In this chapter we will use three methods to solve a system of linear equations. The first method we\u2019ll use is graphing.<\/p>\n<p id=\"fs-id1168341959249\">The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p>\n<p id=\"fs-id1168345511616\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.<\/p>\n<p id=\"fs-id1168345293314\">Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_005\">(Figure 1)<\/a>:<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_05_01_005\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"875\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_005_img_new.jpg\" alt=\"This figure shows three x y-coordinate planes. The first plane shows two lines which intersect at one point. Under the graph it says, \u201cThe lines intersect. Intersecting lines have one point in common. There is one solution to this system.\u201d The second x y-coordinate plane shows two parallel lines. Under the graph it says, \u201cThe lines are parallel. Parallel lines have no points in common. There is no solution to this system.\u201d The third x y-coordinate plane shows one line. Under the graph it says, \u201cBoth equations give the same line. Because we have just one line, there are infinitely many solutions.\u201d\" width=\"875\" height=\"349\" data-media-type=\"image\/jpeg\"> Figure 1[\/caption]\n\n<\/div>\n<p id=\"fs-id1168345325810\">For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we\u2019ll use a different method in each section. After seeing the third method, you\u2019ll decide which method was the most convenient way to solve this system.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How to Solve a System of Linear Equations by Graphing<\/div>\n<div id=\"fs-id1168345283558\" data-type=\"exercise\">\n<div id=\"fs-id1168345451825\" data-type=\"problem\">\n<p id=\"fs-id1168345250621\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345196799\" 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<span id=\"fs-id1168345644826\" data-type=\"media\" data-alt=\"This table has four rows and three columns. The first column acts as the header column. The first row reads, \u201cStep 1. Graph the first equation.\u201d Then it reads, \u201cTo graph the first line, write the equation in slope-intercept form.\u201d The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x \u2013 2y = 6 is also listed.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006a_img_new.jpg\" alt=\"This table has four rows and three columns. The first column acts as the header column. The first row reads, \u201cStep 1. Graph the first equation.\u201d Then it reads, \u201cTo graph the first line, write the equation in slope-intercept form.\u201d The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x \u2013 2y = 6 is also listed.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345191681\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2. Graph the second equation on the same rectangular coordinate system.\u201d Then it says, \u201cTo graph the second line, use intercepts.\u201d This is followed by the equation x \u2013 2y = 6 and the ordered pairs (0, -3) and (6, 0). The last column of this row shows a graph of the two equations.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2. Graph the second equation on the same rectangular coordinate system.\u201d Then it says, \u201cTo graph the second line, use intercepts.\u201d This is followed by the equation x \u2013 2y = 6 and the ordered pairs (0, -3) and (6, 0). The last column of this row shows a graph of the two equations.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345425385\" data-type=\"media\" data-alt=\"The third row reads, \u201cStep 3. Determine whether the lines intersect, are parallel, or are the same line.\u201d Then \u201cLook at the graph of the lines.\u201d Finally it reads, \u201cThe lines intersect.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006c_img_new.jpg\" alt=\"The third row reads, \u201cStep 3. Determine whether the lines intersect, are parallel, or are the same line.\u201d Then \u201cLook at the graph of the lines.\u201d Finally it reads, \u201cThe lines intersect.\u201d\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168341841026\" data-type=\"media\" data-alt=\"The fourth row reads, \u201cStep 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.\u201d Then it reads, \u201cSince the lines intersect, find the point of intersection. Check the point in both equations.\u201d Finally it reads, \u201cThe lines intersect at (4, -1). It then uses substitution to show that, \u201cThe solution is (4, -1).\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006d_img_new.jpg\" alt=\"The fourth row reads, \u201cStep 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.\u201d Then it reads, \u201cSince the lines intersect, find the point of intersection. Check the point in both equations.\u201d Finally it reads, \u201cThe lines intersect at (4, -1). It then uses substitution to show that, \u201cThe solution is (4, -1).\u201d\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345220500\" data-type=\"problem\">\n<p id=\"fs-id1168345431391\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}x-3y=-3\\hfill \\\\ x+y=5\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345560615\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345449012\">\\(\\left(3,2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"font-size: 14pt; text-align: initial;\">The steps to use to solve a system of linear equations by graphing are shown below<\/span>\n<div id=\"fs-id1168345450126\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">To solve a system of linear equations by graphing.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169751856808\" class=\"stepwise\" type=\"1\">\n \t<li>Graph the first equation.<\/li>\n \t<li>Graph the second equation on the same rectangular coordinate system.<\/li>\n \t<li>Determine whether the lines intersect, are parallel, or are the same line.<\/li>\n \t<li>Identify the solution to the system.\n<ul id=\"fs-id1168345443090\" data-bullet-style=\"open-circle\">\n \t<li>If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.<\/li>\n \t<li>If the lines are parallel, the system has no solution.<\/li>\n \t<li>If the lines are the same, the system has an infinite number of solutions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345689915\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345215696\" data-type=\"exercise\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345689915\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345215696\" data-type=\"exercise\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\n<div id=\"fs-id1168345240470\" data-type=\"problem\">\n<p id=\"fs-id1168345450358\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=2x+1\\hfill \\\\ y=4x-1\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345261327\" 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-id1168345360405\">Both of the equations in this system are in slope-intercept form, so we will use their slopes and <em data-effect=\"italics\">y<\/em>-intercepts to graph them. \\(\\left\\{\\begin{array}{c}y=2x+1\\hfill \\\\ y=4x-1\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167829696931\" style=\"width: 100%;\" summary=\"This figure begins with a set of instructions. The first line reads, \u201cFind the slope and y-intercept of the first equation.\u201d Next to this, it shows that y equals 2x plus 1 where m = 2 and b = 1. The next line down reads, \u201cFind the slope and y-intercept of the first equation.\u201d Next to this on the right, it shows that y = 4x \u2013 1 where m = 4 and b = -1. The next line down reads, \u201cGraph the two lines\u201d. Beneath this it reads, \u201cDetermine the point of intersection. The lines intersect at (1, 3). This it shows a graph of the two lines on an x, y coordinate plane. The lines intersect at (1, 3). Then the figure says, \u201cCheck the solution in both equations.\u201d The first equation reads, y = 2 plus 1. Then 3 = 2 times 1 plus 1. Then 3 = 3. The second equation reads, y = 4x \u2013 1. Then 3 = 4 times 1 \u2013 1. Then 3 = 3. Then it says, \u201cThe solution is (1, 3).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\">\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836693016\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\">\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836492140\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the two lines.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-align=\"center\" data-valign=\"top\">The lines intersect at (1, 3).<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833102398\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Check the solution in both equations.<\/td>\n<td data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 2x+1\\hfill \\\\ \\hfill 3&amp; \\stackrel{?}{=}\\hfill &amp; 2\u00b71+1\\hfill \\\\ \\hfill 3&amp; =\\hfill &amp; 3\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 4x-1\\hfill \\\\ \\hfill 3&amp; \\stackrel{?}{=}\\hfill &amp; 4\u00b71-1\\hfill \\\\ \\hfill 3&amp; =\\hfill &amp; 3\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\" data-valign=\"top\">The solution is (1, 3).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\n<p id=\"fs-id1168345270231\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}y=2x+2\\hfill \\\\ y=\\text{\u2212}x-4\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345329125\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345433895\">\\(\\left(-2,-2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345193878\" data-type=\"problem\"><span style=\"orphans: 1; text-align: initial; font-size: 14pt;\">Both equations in Example 3 were given in slope\u2013intercept form. This made it easy for us to quickly graph the lines. In the next example, we\u2019ll first re-write the equations into slope\u2013intercept form.<\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345301935\" data-type=\"problem\">\n<p id=\"fs-id1168345250708\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}3x+y=-1\\hfill \\\\ 2x+y=0\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345302868\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1167836326706\">We\u2019ll solve both of these equations for \\(y\\) so that we can easily graph them using their slopes and <em data-effect=\"italics\">y<\/em>-intercepts. \\(\\left\\{\\begin{array}{c}3x+y=-1\\hfill \\\\ 2x+y=0\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167824584625\" style=\"width: 100%;\" summary=\"This figure shows a solution in two columns. The first line in the left column, reads \u201cSolve the first equation for y.\u201d To the right of this, it has the equation 3x + y = negative 1 which becomes y = negative 3x \u2013 1. The next line down reads, \u201cFind the slope and y-intercept.\u201d Then it shows that m = negative 3 and b = negative 1. Below this reads, \u201cSolve the second equation for y.\u201d Next to this is the equation 2x + 1 = 0 become y = negative 2x. Below this reads, \u201cFind the slope and y-intercept.\u201d Next to this it shows that b = negative 2 and b = 0. Then it says, \u201cGraph the lines.\u201d An x y-coordinate plane shows the two graphed lines intersecting at (negative 1, 2). The figure then indicates, \u201cDetermine the point of intersection. The lines intersect at (negative 1, 2).\u201d Finally the figure says, \u201cCheck the solution in both equations.\u201d The first equation shows 3x + y = -1. Then 3(-1) + 2 = -1. And then -1 = -1. The second equation shows 2x + y = 0. Then 2(-1) + 2 = 0. Then 0 = 0. The figure then says, \u201cThe solution is (-1, 2).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Solve the second equation for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{c}\\begin{array}{ccc}\\hfill 3x+y&amp; =\\hfill &amp; -1\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -3x-1\\hfill \\\\ \\\\ \\hfill m&amp; =\\hfill &amp; -3\\hfill \\\\ \\hfill b&amp; =\\hfill &amp; -1\\hfill \\\\ \\\\ \\\\ \\hfill 2x+y&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -2x\\hfill \\\\ \\\\ \\hfill m&amp; =\\hfill &amp; -2\\hfill \\\\ \\hfill b&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill \\end{array}\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836705596\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-align=\"center\" data-valign=\"top\">The lines intersect at (\u22121, 2).<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Check the solution in both equations.<\/td>\n<td data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill 3x+y&amp; =\\hfill &amp; -1\\hfill \\\\ \\hfill 3\\left(-1\\right)+2&amp; \\stackrel{?}{=}\\hfill &amp; -1\\hfill \\\\ \\hfill -1&amp; =\\hfill &amp; -1\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill 2x+y&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill 2\\left(-1\\right)+2&amp; \\stackrel{?}{=}\\hfill &amp; 0\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 0\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\" data-valign=\"top\">The solution is (\u22121, 2).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345337170\" data-type=\"problem\">\n<p id=\"fs-id1168345301261\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}-x+y=1\\hfill \\\\ 2x+y=10\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345561188\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345254462\">\\(\\left(3,4\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345398741\">Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We\u2019ll do this in the next example.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345644007\" data-type=\"problem\">\n<p id=\"fs-id1168345695461\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x+y=2\\hfill \\\\ x-y=4\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345273530\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345276953\">We will find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts of both equations and use them to graph the lines.<\/p>\n\n<table id=\"fs-id1167829578991\" style=\"width: 100%;\" summary=\"This is a figure that shows a solution in three columns. In the first column, it reads, \u201cTo find the intercepts, let x = 0 and solve for y, then let y = 0 and solve for x.\u201d The middle column shows the equation x + y = 2. Centered below this it reads x + y = 2 and next to this, x + y = 2. Below this it shows 0 + y = 2, and y = 2. Then it shows x + 0 = 2, and x = 2. There is a table to the right with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the number 0 and 2. Under \u201cy\u201d are the numbers 2 and 0. In the next row down, it reads, \u201cFind the intercepts, let x = 0 then let y = 0.\u201d The middle column shows x - y = 4. Then x - y = 4 and x - y = 4. Below this it shows 0 - y = 4, -y = 4, and y = negative 4. Then it shows x - 0 = 4, and x = 4. There is a table to the right with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the number 0 and 4. Under \u201cy\u201d are the numbers -4 and 0. Below this there is a graph that shows two lines intersecting at point (3, -1) on an x y-coordinate plane.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836629538\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">To find the intercepts, let <em data-effect=\"italics\">x<\/em> = 0 and solve<span data-type=\"newline\">\n<\/span>for <em data-effect=\"italics\">y<\/em>, then let <em data-effect=\"italics\">y<\/em> = 0 and solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill x+y&amp; =\\hfill &amp; 2\\hfill \\\\ 0+y&amp; =\\hfill &amp; 2\\hfill \\\\ y&amp; =\\hfill &amp; 2\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill x+y&amp; =\\hfill &amp; 2\\hfill \\\\ x+0&amp; =\\hfill &amp; 2\\hfill \\\\ x&amp; =\\hfill &amp; 2\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><span id=\"fs-id1167829596067\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836621931\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">To find the intercepts, let<span data-type=\"newline\">\n<\/span><em data-effect=\"italics\">x<\/em> = 0 then let <em data-effect=\"italics\">y<\/em> = 0.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill x-y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill 0-y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill -y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -4\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill x-y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill x-0&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 4\\hfill \\\\ \\\\ \\\\ \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829830534\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_010b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Graph the line.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836320119\" data-type=\"media\" data-alt=\"This graph shows two lines intersection at point (3, -1) on an x y-coordinate plane.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_011_img_new.jpg\" alt=\"This graph shows two lines intersection at point (3, -1) on an x y-coordinate plane.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Determine the point of intersection.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\">The lines intersect at (3, \u22121).<\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Check the solution in both equations.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccccccc}\\hfill x+y&amp; =\\hfill &amp; 2\\hfill &amp; &amp; &amp; \\hfill x-y&amp; =\\hfill &amp; 4\\hfill \\\\ 3+\\left(-1\\right)\\hfill &amp; \\stackrel{?}{=}\\hfill &amp; 2\\hfill &amp; &amp; &amp; \\hfill 3-\\left(-1\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ \\hfill 2&amp; =\\hfill &amp; 2\u2713\\hfill &amp; &amp; &amp; \\hfill 4&amp; =\\hfill &amp; 4\u2713\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span>The solution is (3, \u22121).<\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345727147\" data-type=\"problem\">\n<p id=\"fs-id1168345301947\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}x+y=6\\hfill \\\\ x-y=2\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345292167\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345390050\">\\(\\left(4,2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345389141\">Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345431017\" data-type=\"problem\">\n<p id=\"fs-id1168345419137\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=6\\hfill \\\\ 2x+3y=12\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345287687\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836356457\" style=\"width: 100%;\" summary=\"This figure shows two equations: y = 6 and x plus 3y = 12. It then says, \u201cWe know the first equation represents a horizontal line whose y-intercept is 6.\u201d Then is shows that y = 6. It then says, \u201cThe second equation is most conveniently graphed using intercepts.\u201d Then it shows 2x plus 3y = 12. Then it reads, \u201cTo find the intercepts, let x = 0 and then y = 0.\u201d There is a graph with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and 6. Under \u201cy\u201d are the numbers 4 and 0. Then it reads, \u201cGraph the lines.\u201d The two lines are graphed on an x y-coordinate plane. The lines intersect at point (negative 3, 6). The figure then reads, \u201cDetermine the point of intersection. The lines intersect at (negative 3, 6).\u201d The figure then shows that y = 6, then 6 = 6, and 2 = 2. It also shows that 2x +3y = 12. Then, 2 times negative 3, in parentheses, plus 3 times 6 = 12. Then negative 6 plus 18 = 12, and 12 = 12. The figure then indicates, \u201cThe solution is (negative 3, 6).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 24.2045%;\"><\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167829594693\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">We know the first equation represents a horizontal<span data-type=\"newline\">\n<\/span>line whose <em data-effect=\"italics\">y<\/em>-intercept is 6.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836550964\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">The second equation is most conveniently graphed<span data-type=\"newline\">\n<\/span>using intercepts.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836492417\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">To find the intercepts, let <em data-effect=\"italics\">x<\/em> = 0 and then <em data-effect=\"italics\">y<\/em> = 0.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836355778\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Graph the lines.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167829790109\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Determine the point of intersection.<\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\">The lines intersect at (\u22123, 6).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Check the solution to both equations.<\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\">\\(\\begin{array}{cccccccc}\\hfill y&amp; =\\hfill &amp; 6\\hfill &amp; &amp; &amp; \\hfill 2x+3y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill 6&amp; \\stackrel{?}{=}\\hfill &amp; 6\u2713\\hfill &amp; &amp; &amp; \\hfill 2\\left(-3\\right)+3\\left(6\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 12\\hfill \\\\ \\hfill 2&amp; =\\hfill &amp; 2\\hfill &amp; &amp; &amp; \\hfill -6+18&amp; \\stackrel{?}{=}\\hfill &amp; 12\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill 12&amp; =\\hfill &amp; 12\u2713\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\"><\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\">The solution is (\u22123, 6).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345250431\" data-type=\"problem\">\n<p id=\"fs-id1168345230324\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}y=-1\\hfill \\\\ x+3y=6\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345229034\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345286142\">\\(\\left(9,-1\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"font-size: 14pt; text-align: initial;\">In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we\u2019ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.<\/span>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341955833\" data-type=\"problem\">\n<p id=\"fs-id1168341864012\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=\\frac{1}{2}x-3\\hfill \\\\ x-2y=4\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345260776\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836599119\" style=\"width: 100%;\" summary=\"This figure lists two equations: y = one half \u201cx\u201d minus 3 and x minus 2 = 4. It then indicates, \u201cTo graph the first equation, we will use the slope and y-intercept.\u201d The figure then shows that y = one half \u201cx\u201d minus 3 in which m = one half and b = negative 3. The next line down reads, \u201cTo graph the second equation, we will use the intercepts.\u201d Next to this is the equation x minus 2y = 4 which is followed by a table with two columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and 4. Under \u201cy\u201d are the numbers negative 2 and 0. The figure then says, \u201cGraph the lines.\u201d The two lines are then graphed on an x y-coordinate plane. The two lines appear to be parallel. The figure then reads, \u201cDetermine the point of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167825829989\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">To graph the first equation, we will<span data-type=\"newline\">\n<\/span>use its slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829877876\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829753121\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836557401\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">To graph the second equation,<span data-type=\"newline\">\n<\/span>we will use the intercepts.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836543355\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833022880\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836366683\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-valign=\"top\">\u2003\u2003\u2003\u2003The lines are parallel.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Since no point is on both lines, there is no ordered pair<span data-type=\"newline\">\n<\/span>that makes both equations true. There is no solution to<span data-type=\"newline\">\n<\/span>this system.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345746509\" data-type=\"problem\">\n<p id=\"fs-id1168345420434\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}y=-\\frac{1}{4}x+2\\hfill \\\\ x+4y=-8\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345725138\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345418010\">no solution<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916918\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168341913823\" data-type=\"exercise\">\n<div id=\"fs-id1168345746509\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345194746\" data-type=\"problem\">\n<p id=\"fs-id1168345428978\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=2x-3\\hfill \\\\ -6x+3y=-9\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345510019\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836510916\" style=\"width: 100%;\" summary=\"This figure shows two equations: y = 2x minus 3 and negative 6x plus 3y = negative 9. The next line down reads, \u201cFind the slope and y-intercept of the first equation.\u201d It then shows that y = 2x -3 where m = 2 and b = negative 3. I also shows negative 6x plus 3y = negative 9. It then reads, \u201cFind the intercepts of the second equation.\u201d There is a table with two columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and three over two. Under \u201cy\u201d are the numbers negative 3 and 0. The figure then reads, \u201cGraph the lines.\u201d There appears to be only one line graphed on the x y coordinate plane. The figure then reads, \u201cDetermine the point of intersection. The lines are the same! Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to this system.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833380846\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\">\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836628539\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the intercepts of the second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829599268\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833139693\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836287817\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-valign=\"top\">The lines are the same!<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Since every point on the line makes both equations<span data-type=\"newline\">\n<\/span>true, there are infinitely many ordered pairs that make<span data-type=\"newline\">\n<\/span>both equations true.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">There are infinitely many solutions to this system.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345414817\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345650454\" data-type=\"exercise\">\n<div id=\"fs-id1168345302969\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345303045\" data-type=\"problem\">\n<p id=\"fs-id1168345445689\">Solve each system by graphing: \\(\\left\\{\\begin{array}{c}y=-3x-6\\hfill \\\\ 6x+2y=-12\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345407885\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345453131\">infinitely many solutions<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"font-size: 14pt; text-align: initial;\">If you write the second equation in Example 8 in slope-intercept form, you may recognize that the equations have the same slope and same <\/span><em style=\"font-size: 14pt; text-align: initial;\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt; text-align: initial;\">-intercept.<\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345329224\">When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Coincident Lines<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nCoincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341840730\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Determine the Number of Solutions of a Linear System<\/h1>\n<p id=\"fs-id1168345262002\">There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.<\/p>\n<p id=\"fs-id1168345325732\">We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example 2 through Example 6 all had two intersecting lines. Each system had one solution.<\/p>\n<p id=\"fs-id1168345230044\">A system with parallel lines, like Example 7, has no solution. What happened in Example 8? The equations have <span class=\"no-emphasis\" data-type=\"term\">coincident lines<\/span>, and so the system had infinitely many solutions.<\/p>\n<p id=\"fs-id1168345237685\">We\u2019ll organize these results in <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_018_img\">(Table 1)<\/a> below:<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_05_01_018_img\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"317\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_018_img_new.jpg\" alt=\"This table has two columns and four rows. The first row labels each column \u201cGraph\u201d and \u201cNumber of solutions.\u201d Under \u201cGraph\u201d are \u201c2 intersecting lines,\u201d \u201cParallel lines,\u201d and \u201cSame line.\u201d Under \u201cNumber of solutions\u201d are \u201c1,\u201d \u201cNone,\u201d and \u201cInfinitely many.\u201d\" width=\"317\" height=\"121\" data-media-type=\"image\/jpeg\"> Table 1[\/caption]\n\n<\/div>\n<p id=\"fs-id1168345693726\">Parallel lines have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. So, if we write both equations in a system of linear equations in slope\u2013intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 7.<\/p>\n\n<div id=\"fs-id1168345743284\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{0.1em}{0ex}}\\left\\{\\phantom{\\rule{0.1em}{0ex}}\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; \\frac{1}{2}x-3\\hfill \\\\ \\hfill x-2y&amp; =\\hfill &amp; 4\\hfill \\end{array}\\hfill \\\\ \\text{The first line is in slope\u2013intercept form.}\\hfill &amp; &amp; &amp; \\text{If we solve the second equation for}\\phantom{\\rule{0.2em}{0ex}}y,\\phantom{\\rule{0.2em}{0ex}}\\text{we get}\\hfill \\\\ \\hfill y=\\frac{1}{2}x-3\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\begin{array}{ccc}\\hfill x-2y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill -2y&amp; =\\hfill &amp; \\text{\u2212}x+4\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; \\frac{1}{2}x-2\\hfill \\end{array}\\hfill \\\\ \\hfill m=\\frac{1}{2},b=-3\\hfill &amp; &amp; &amp; \\hfill m=\\frac{1}{2},b=-2\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345287565\">The two lines have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. They are parallel lines.<\/p>\n<p id=\"fs-id1168345453187\"><a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_019_img\">(Table 2)<\/a> shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_05_01_019_img\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"486\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_019_img_new.jpg\" alt=\"This table is entitled \u201cNumber of Solutions of a Linear System of Equations.\u201d There are four columns. The columns are labeled, \u201cSlopes,\u201d \u201cIntercepts,\u201d \u201cType of Lines,\u201d \u201cNumber of Solutions.\u201d Under \u201cSlopes\u201d are \u201cDifferent,\u201d \u201cSame,\u201d and \u201cSame.\u201d Under \u201cIntercepts,\u201d the first cell is blank, then the words \u201cDifferent\u201d and \u201cSame\u201d appear. Under \u201cTypes of Lines\u201d are the words, \u201cIntersecting,\u201d \u201cParallel,\u201d and \u201cCoincident.\u201d Under \u201cNumber of Solutions\u201d are \u201c1 point,\u201d \u201cNo Solution,\u201d and \u201cInfinitely many solutions.\u201d\" width=\"486\" height=\"148\" data-media-type=\"image\/jpeg\"> Table 2[\/caption]\n\n<\/div>\n<p id=\"fs-id1168345287740\">Let\u2019s take one more look at our equations in <a class=\"autogenerated-content\" href=\"#fs-id1168341955833\">(Example 7)<\/a> that gave us parallel lines.<\/p>\n\n<div id=\"fs-id1168345510136\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}y=\\frac{1}{2}x-3\\hfill \\\\ x-2y=4\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345650553\">When both lines were in slope-intercept form we had:<\/p>\n\n<div id=\"fs-id1168345302936\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(y=\\frac{1}{2}x-3\\phantom{\\rule{2em}{0ex}}y=\\frac{1}{2}x-2\\)<\/div>\n<p id=\"fs-id1168345457988\">Do you recognize that it is impossible to have a single ordered pair \\(\\left(x,y\\right)\\) that is a solution to both of those equations?<\/p>\n<p id=\"fs-id1168345287296\">We call a system of equations like this <span class=\"no-emphasis\" data-type=\"term\">an inconsistent system<\/span>. It has no solution.<\/p>\n<p id=\"fs-id1168345692186\">A system of equations that has at least one solution is called a <span class=\"no-emphasis\" data-type=\"term\">consistent system<\/span>.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Consistent and Inconsistent Systems<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1168345692954\">A <span data-type=\"term\">consistent system<\/span> of equations is a system of equations with at least one solution.<\/p>\n<p id=\"fs-id1168345427031\">An <span data-type=\"term\">inconsistent system<\/span> of equations is a system of equations with no solution.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1168345240009\">We also categorize the equations in a system of equations by calling the equations <em data-effect=\"italics\">independent<\/em> or <em data-effect=\"italics\">dependent<\/em>. If two equations are <span data-type=\"term\">independent equations<\/span>, they each have their own set of solutions. Intersecting lines and parallel lines are independent.<\/p>\n<p id=\"fs-id1168345228631\">If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two <span class=\"no-emphasis\" data-type=\"term\">dependent equations<\/span>, we get coincident lines.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Independent and Dependent Equations<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1168345423433\">Two equations are <span data-type=\"term\">independent<\/span> if they have different solutions.<\/p>\n<p id=\"fs-id1168345278054\">Two equations are <span data-type=\"term\">dependent<\/span> if all the solutions of one equation are also solutions of the other equation.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1168345417086\">Let\u2019s sum this up by looking at the graphs of the three types of systems. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_015\">(Figure 3)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_020_img\">(Table 3)<\/a>.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_05_01_015\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"791\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_015_img_new.jpg\" alt=\"This figure shows three x y coordinate planes in a horizontal row. The first shows two lines intersecting. The second shows two parallel lines. The third shows two coincident lines.\" width=\"791\" height=\"257\" data-media-type=\"image\/jpeg\"> Figure 3[\/caption]\n\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_05_01_020_img\" class=\"bc-figure figure\"><\/div>\n<div class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"528\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_020_img_new.jpg\" alt=\"This table has four columns and four rows. The columns are labeled, \u201cLines,\u201d \u201cIntersecting,\u201d \u201cParallel,\u201d and \u201cCoincident.\u201d In the first row under the labeled column \u201clines\u201d it reads \u201cNumber of solutions.\u201d Reading across, it tell us that an intersecting line contains 1 point, a parallel line provides no solution, and a coincident line has infinitely many solutions. A consistent\/inconsistent line has consistent lines if they are intersecting, inconsistent lines if they are parallel and consistent if the lines are coincident. Finally, dependent and independent lines are considered independent if the lines intersect, they are also independent if the lines are parallel, and they are dependent if the lines are coincident.\" width=\"528\" height=\"122\" data-media-type=\"image\/jpeg\"> Table 3[\/caption]\n\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345448163\" data-type=\"problem\">\n<p id=\"fs-id1168345671375\">Without graphing, determine the number of solutions and then classify the system of equations: \\(\\left\\{\\begin{array}{c}y=3x-1\\hfill \\\\ 6x-2y=12\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345251139\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-118\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We will compare the slopes and intercepts of the two lines.<\/td>\n<td>\\(\\left\\{\\begin{array}{c}y=3x-1\\hfill \\\\ 6x-2y=12\\hfill \\end{array}.\\)<\/td>\n<\/tr>\n<tr>\n<td>The first equation is already in slope-intercept form.<\/td>\n<td>\\(y=3x-1\\)<\/td>\n<\/tr>\n<tr>\n<td>Write the second equation in slope-intercept form.<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill 6x-2y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill -2y&amp; =\\hfill &amp; -6x+12\\hfill \\\\ \\hfill \\frac{-2y}{-2}&amp; =\\hfill &amp; \\frac{-6x+12}{-2}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; 3x-6\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Find the slope and intercept of each line.<\/td>\n<td>\\(\\begin{array}{ccccccccc}\\hfill y&amp; =\\hfill &amp; 3x-1\\hfill &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; 3x-6\\hfill \\\\ \\hfill m&amp; =\\hfill &amp; 3\\hfill &amp; &amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; 3\\hfill \\\\ \\hfill b&amp; =\\hfill &amp; -1\\hfill &amp; &amp; &amp; &amp; \\hfill b&amp; =\\hfill &amp; -6\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>Since the slopes are the same and \\(y\\)-intercepts are different, the lines are parallel.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345560002\">A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345326624\" data-type=\"problem\">\n<p id=\"fs-id1168345416501\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345224525\">\\(\\left\\{\\begin{array}{c}y=-2x-4\\hfill \\\\ 4x+2y=9\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345415558\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345667907\">no solution, inconsistent, independent<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341840730\" class=\"bc-section section\" data-depth=\"1\">\n<div id=\"fs-id1168345538684\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345408145\" data-type=\"exercise\">\n<div id=\"fs-id1168345414201\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341857668\" data-type=\"problem\">\n<p id=\"fs-id1168341857670\">Without graphing, determine the number of solutions and then classify the system of equations: \\(\\left\\{\\begin{array}{c}2x+y=-3\\hfill \\\\ x-5y=5\\hfill \\end{array}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345553490\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-504\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We will compare the slope and intercepts of the two lines.<\/td>\n<td colspan=\"2\" data-align=\"center\">\\(\\left\\{\\begin{array}{c}2x+y=-3\\hfill \\\\ x-5y=5\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Write both equations in slope-intercept form.<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill 2x+y&amp; =&amp; -3\\hfill \\\\ \\hfill y&amp; =&amp; -2x-3\\hfill \\end{array}\\)<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill x-5y=5&amp; =&amp; 5\\hfill \\\\ \\hfill -5y&amp; =&amp; -x+5\\hfill \\\\ \\hfill \\frac{-5y}{-5}&amp; =&amp; \\frac{-x+5}{-5}\\hfill \\\\ \\hfill y&amp; =&amp; \\frac{1}{5}x-1\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Find the slope and intercept of each line.<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill y&amp; =&amp; -2x-3\\hfill \\\\ \\hfill m&amp; =&amp; -2\\hfill \\\\ \\hfill b&amp; =&amp; -3\\hfill \\end{array}\\)<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill y&amp; =&amp; \\frac{1}{5}x-1\\hfill \\\\ \\hfill m&amp; =&amp; \\frac{1}{5}\\hfill \\\\ \\hfill b&amp; =&amp; -1\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td colspan=\"2\">Since the slopes are different, the lines intersect.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345742349\">A system of equations whose graphs are intersect has 1 solution and is consistent and independent.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345417929\" data-type=\"problem\">\n<p id=\"fs-id1168345398213\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345398216\">\\(\\left\\{\\begin{array}{c}3x+2y=2\\hfill \\\\ 2x+y=1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345458749\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345458751\">one solution, consistent, independent<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345425501\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345417927\" data-type=\"exercise\">\n<div id=\"fs-id1168345417929\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345557327\" data-type=\"problem\">\n<p id=\"fs-id1168345557329\">Without graphing, determine the number of solutions and then classify the system of equations. \\(\\left\\{\\begin{array}{c}3x-2y=4\\hfill \\\\ y=\\frac{3}{2}x-2\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345384539\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-238\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We will compare the slopes and intercepts of the two lines.<\/td>\n<td>\\(\\left\\{\\begin{array}{c}3x-2y=4\\hfill \\\\ y=\\frac{3}{2}x-2\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Write the first equation in slope-intercept form.<\/td>\n<td>\\(\\begin{array}{ccc}\\hfill 3x-2y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill -2y&amp; =\\hfill &amp; -3x+4\\hfill \\\\ \\hfill \\frac{-2y}{-2}&amp; =\\hfill &amp; \\frac{-3x+4}{-2}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; \\frac{3}{2}x-2\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>The second equation is already in slope-intercept form.<\/td>\n<td>\\(\\begin{array}{c}\\hfill y=\\frac{3}{2}x-2\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>Since the slopes are the same, they have the same slope and same \\(y\\)-intercept and so the lines are coincident.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168341852918\">A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345458749\" data-type=\"solution\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345374488\" data-type=\"problem\">\n<p id=\"fs-id1168345374490\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345398366\">\\(\\left\\{\\begin{array}{c}4x-5y=20\\hfill \\\\ y=\\frac{4}{5}x-4\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345428781\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345428783\">infinitely many solutions, consistent, dependent<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916335\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Graphing<\/h1>\n<p id=\"fs-id1168345743034\">We will modify the\u00a0 problem solving strategy slightly to set up and solve applications of systems of linear equations.<\/p>\n\n<div id=\"fs-id1168341972806\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">How to use a problem solving strategy for systems of linear equations.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341972806\" class=\"howto\" data-type=\"note\">\n<ol id=\"fs-id1169751898179\" class=\"stepwise\" type=\"1\">\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n \t<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1168341910497\">Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345415363\" data-type=\"problem\">\n<p id=\"fs-id1168345415365\">Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345667515\" 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-id1168341961495\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/p>\n<p id=\"fs-id1168341841009\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/p>\n<p id=\"fs-id1168345447831\">We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.<\/p>\n<p id=\"fs-id1168345447835\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/p>\n<p id=\"fs-id1168037092275\">\u2003\u2003Let \\(f=\\) number of quarts of fruit juice.<span data-type=\"newline\">\n<\/span> \u2003\u2003\u2003\u2003\\(c=\\) number of quarts of club soda<\/p>\n<p id=\"fs-id1168341958682\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<span data-type=\"newline\">\n<\/span><\/p>\n<span id=\"fs-id1168341917919\" data-type=\"media\" data-alt=\"This figure shows sentences converted into equations. The first sentence reads, \u201cThe number of quarts of fruit juice and the number of quarts of club soda is 10. \u201cNumber of quarts of fruit juice\u201d contains a curly bracket beneath the phrase with an \u201cf\u201d centered under the bracket. The \u201cAnd\u201d also contains a curly bracket beneath it and has a plus sign centered beneath it. \u201cNumber of quarts of club soda\u201d contains a curly bracket with the variable \u201cc\u201d beneath it. And finally, the phrase \u201cis 10\u201d contains a curly bracket. Under this it reads equals 10. The second sentence reads, \u201cThe number of quarts of fruit juice is four times the number of quarts of club soda\u201d. This sentence is set up similarly in that each phrase contains a curly bracket underneath. The variable \u201cf\u201d represents \u201cThe number of quarts of fruit juice\u201d. An equal sign represents \u201cis\u201d and \u201c4c\u201d represents four times the number of quarts of club soda.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_016_img_new.jpg\" alt=\"This figure shows sentences converted into equations. The first sentence reads, \u201cThe number of quarts of fruit juice and the number of quarts of club soda is 10. \u201cNumber of quarts of fruit juice\u201d contains a curly bracket beneath the phrase with an \u201cf\u201d centered under the bracket. The \u201cAnd\u201d also contains a curly bracket beneath it and has a plus sign centered beneath it. \u201cNumber of quarts of club soda\u201d contains a curly bracket with the variable \u201cc\u201d beneath it. And finally, the phrase \u201cis 10\u201d contains a curly bracket. Under this it reads equals 10. The second sentence reads, \u201cThe number of quarts of fruit juice is four times the number of quarts of club soda\u201d. This sentence is set up similarly in that each phrase contains a curly bracket underneath. The variable \u201cf\u201d represents \u201cThe number of quarts of fruit juice\u201d. An equal sign represents \u201cis\u201d and \u201c4c\u201d represents four times the number of quarts of club soda.\u201d\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345374822\">We now have the system. \\(\\left\\{\\begin{array}{c}f+c=10\\hfill \\\\ f=4c\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1168345558568\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations using good algebra techniques.<span data-type=\"newline\">\n<\/span><\/p>\n<span id=\"fs-id1168345634396\" data-type=\"media\" data-alt=\"This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_017_img_new.jpg\" alt=\"This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345386283\">The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.<\/p>\n<p id=\"fs-id1168345329449\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/p>\n<p id=\"fs-id1168345342682\">Does this make sense in the problem?<\/p>\n<p id=\"fs-id1168345342685\">Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.<\/p>\n<p id=\"fs-id1168345457793\">Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.<\/p>\n<p id=\"fs-id1168345457796\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question with a complete sentence.<\/p>\n<p id=\"fs-id1168345425413\">Sondra needs 8 quarts of fruit juice and 2 quarts of soda.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345461354\" data-type=\"problem\">\n<p id=\"fs-id1168345461356\">Manu is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manu need?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345748496\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345748498\">Manu needs 3 quarts juice concentrate and 9 quarts water.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341952013\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1168345745168\">Access these online resources for additional instruction and practice with solving systems of equations by graphing.<\/p>\n\n<ul id=\"fs-id1168345745172\" data-display=\"block\">\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25linsysGraph\">Instructional Video Solving Linear Systems by Graphing<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25solvesbyGraph\">Instructional Video Solve by Graphing<\/a><\/li>\n<\/ul>\n<h1>Key Concepts<\/h1>\n<ul>\n \t<li><strong data-effect=\"bold\">Problem Solving Strategy for Systems of Linear Equations<\/strong>\n<ol id=\"fs-id1169754134591\" class=\"stepwise\" type=\"1\">\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n \t<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1 data-type=\"glossary-title\">Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1168345631530\">\n \t<dt>coincident lines<\/dt>\n \t<dd id=\"fs-id1168345527779\">Coincident lines are lines that have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345527788\">\n \t<dt>consistent system<\/dt>\n \t<dd id=\"fs-id1168345527794\">A consistent system of equations is a system of equations with at least one solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345527798\">\n \t<dt>dependent equations<\/dt>\n \t<dd id=\"fs-id1168345484131\">Two equations are dependent if all the solutions of one equation are also solutions of the other equation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484135\">\n \t<dt>inconsistent system<\/dt>\n \t<dd id=\"fs-id1168345484140\">An inconsistent system of equations is a system of equations with no solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484145\">\n \t<dt>independent equations<\/dt>\n \t<dd id=\"fs-id1168345484150\">Two equations are independent if they have different solutions.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484154\">\n \t<dt>solutions of a system of equations<\/dt>\n \t<dd id=\"fs-id1168345693483\">Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>).<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345693499\">\n \t<dt>system of linear equations<\/dt>\n \t<dd id=\"fs-id1168345363996\">When two or more linear equations are grouped together, they form a system of linear equations.<\/dd>\n<\/dl>\n<\/div>\n<h1 data-type=\"glossary-title\"><span style=\"font-size: 1.2em; word-spacing: normal;\">4.1 Exercise Set<\/span><\/h1>\n<div id=\"fs-id1168345433887\" class=\"practice-perfect\" data-depth=\"2\">\n<p id=\"fs-id1168345500508\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\\(\\left\\{\\begin{array}{c}2x-6y=0\\hfill \\\\ 3x-4y=5\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>\\(\\left(3,1\\right)\\)<\/li>\n \t<li>\\(\\left(-3,4\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=5\\hfill \\\\ x+y=1\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>\\(\\left(4,\\text{\u22123}\\right)\\)<\/li>\n \t<li>\\(\\left(2,0\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=2\\hfill \\\\ y=\\frac{3}{4}x\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>\\(\\left(\\frac{8}{7},\\frac{6}{7}\\right)\\)<\/li>\n \t<li>\\(\\left(1,\\frac{3}{4}\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+5y=10\\hfill \\\\ y=\\frac{3}{5}x+1\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>\\(\\left(-10,4\\right)\\)<\/li>\n \t<li>(\\left(\\frac{5}{4},\\frac{7}{4}\\right)\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1168345452960\">In the following exercises, solve the following systems of equations by graphing.<\/p>\n\n<ol class=\"twocolumn\" start=\"5\">\n \t<li>\\(\\left\\{\\begin{array}{c}3x+y=-3\\hfill \\\\ 2x+3y=5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-3x+y=-1\\hfill \\\\ 2x+y=4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=x+2\\hfill \\\\ y=-2x+2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=\\frac{3}{2}x+1\\hfill \\\\ y=-\\frac{1}{2}x+5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-x+y=-3\\hfill \\\\ 4x+4y=4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-3x+y=-1\\hfill \\\\ 2x+y=4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=5\\hfill \\\\ 2x-y=4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=2\\hfill \\\\ x-y=0\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=-5\\hfill \\\\ x-y=3\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=-4\\hfill \\\\ -x+2y=-2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-2x+3y=3\\hfill \\\\ x+3y=12\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+3y=6\\hfill \\\\ y=-2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x-3y=-3\\hfill \\\\ y=2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x-y=-1\\hfill \\\\ x=1\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x-3y=-6\\hfill \\\\ x=-3\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}4x-3y=8\\hfill \\\\ 8x-6y=14\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-2x+4y=4\\hfill \\\\ y=\\frac{1}{2}x\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x=-3y+4\\hfill \\\\ 2x+6y=8\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=6\\hfill \\\\ -8x-4y=-24\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+3y=-6\\hfill \\\\ 4y=-\\frac{4}{3}x-8\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-3x+2y=-2\\hfill \\\\ y=\\text{\u2212}x+4\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<div id=\"fs-id1168345521355\" data-type=\"exercise\">\n<div id=\"fs-id1168345521357\" data-type=\"problem\">\n<p id=\"fs-id1168345521359\"><span style=\"text-align: initial; font-size: 14pt;\">Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.<\/span><\/p>\n\n<ol class=\"twocolumn\" start=\"26\">\n \t<li>\\(\\left\\{\\begin{array}{c}y=\\frac{2}{3}x+1\\hfill \\\\ -2x+3y=5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=-2x+1\\hfill \\\\ 4x+2y=8\\hfill \\end{array}\\)<\/li>\n \t<li>missing<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}4x+2y=10\\hfill \\\\ 4x-2y=-6\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=-\\frac{1}{2}x+5\\hfill \\\\ x+2y=10\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=2x+3\\hfill \\\\ 2x-y=-3\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345557294\" data-type=\"exercise\">\n<div id=\"fs-id1168345557296\" data-type=\"problem\">\n<p id=\"fs-id1168345557298\"><span style=\"text-align: initial; font-size: 14pt;\">In the following exercises, solve.<\/span><\/p>\n\n<ol start=\"32\">\n \t<li>Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?<\/li>\n \t<li>Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?<\/li>\n \t<li>Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345572431\" class=\"writing\" data-depth=\"2\">\n<div id=\"fs-id1168341864103\" data-type=\"exercise\">\n<div id=\"fs-id1168341864105\" data-type=\"problem\">\n<h1>Answers<\/h1>\n<ol class=\"twocolumn\">\n \t<li><span style=\"font-family: inherit; font-size: inherit; background-color: initial;\">a) yes b) <\/span><span style=\"font-family: inherit; font-size: inherit; background-color: initial;\">no<\/span><\/li>\n \t<li><span class=\"token\">a)<\/span> yes b) no<\/li>\n \t<li><span class=\"token\">a)<\/span> yes b) no<\/li>\n \t<li><span class=\"token\">a)<\/span> no b) yes<\/li>\n \t<li>\\(\\left(-2,3\\right)\\)<\/li>\n \t<li>\\(\\left(1,2\\right)\\)<\/li>\n \t<li>\\(\\left(0,2\\right)\\)<\/li>\n \t<li>\\(\\left(2,4\\right)\\)<\/li>\n \t<li>\\(\\left(2,-1\\right)\\)<\/li>\n \t<li>\\(\\left(1,2\\right)\\)<\/li>\n \t<li>\\(\\left(3,2\\right)\\)<\/li>\n \t<li>\\(\\left(1,1\\right)\\)<\/li>\n \t<li>\\(\\left(-1,-4\\right)\\)<\/li>\n \t<li>\\(\\left(3,3\\right)\\)<\/li>\n \t<li>\\(\\left(-5,6\\right)\\)<\/li>\n \t<li>\\(\\left(6,-2\\right)\\)<\/li>\n \t<li>\\(\\left(3,2\\right)\\)<\/li>\n \t<li>\\(\\left(1,3\\right)\\)<\/li>\n \t<li>\\(\\left(-3,1\\right)\\)<\/li>\n \t<li>no solution<\/li>\n \t<li>no solution<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=6\\hfill \\\\ -8x-4y=-24\\hfill \\end{array}\\)<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>\\(\\left(2,2\\right)\\)<\/li>\n \t<li>no solutions<\/li>\n \t<li>no solutions<\/li>\n \t<li>no solutions, inconsistent, independent<\/li>\n \t<li>consistent, 1 solution<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>Molly needs 16 ounces of strawberry juice and 48 ounces of water.<\/li>\n \t<li>Enrique needs 8 ounces of nuts and 16 ounces of water.<\/li>\n \t<li>Leo should plant 50 tulips and 300 daffodils.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>","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<ul>\n<li>Determine whether an ordered pair is a solution of a system of equations<\/li>\n<li>Solve a system of linear equations by graphing<\/li>\n<li>Determine the number of solutions of linear system<\/li>\n<li>Solve applications of systems of equations by graphing<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Determine Whether an Ordered Pair is a Solution of a System of Equations<\/h1>\n<div id=\"fs-id1168341923821\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345398220\">\u00a0We learned before how to solve linear equations with one variable. Now we will work with <span data-type=\"term\">systems of linear equations<\/span>, two or more linear equations grouped together, witch is known as a system of linear equations.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">System of Linear Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>When two or more linear equations are grouped together, they form a system of linear equations.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345269961\">We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.<\/p>\n<p id=\"fs-id1168341857638\">An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.<\/p>\n<div id=\"fs-id1168345242279\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-67d56114ed51dc4537e8df2392156c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168345435392\">A linear equation in two variables, like 2<em data-effect=\"italics\">x<\/em> + <em data-effect=\"italics\">y<\/em> = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.<\/p>\n<p id=\"fs-id1168345385750\">To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>) that make both equations true. These are called the <span class=\"no-emphasis\" data-type=\"term\">solutions to a system of equations<\/span>.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Solutions of a System of Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong data-effect=\"bold\">Solutions of a system of equations<\/strong> are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>).<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345430694\">To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.<\/p>\n<p id=\"fs-id1168345561194\">Let\u2019s consider the system below:<\/p>\n<div id=\"fs-id1168345250439\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5f45f572bb1c8b17c9f86942edb5ef63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168345452720\">Is the ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-25c4864d7eaa7ae5b2fe81ae29cf46af_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> a solution?<\/p>\n<p><span id=\"fs-id1168345451927\" data-type=\"media\" data-alt=\"This figure begins with a sentence, \u201cWe substitute x =2 and y = -1 into both equations.\u201d The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_01_001_img_new.jpg\" alt=\"This figure begins with a sentence, \u201cWe substitute x =2 and y = -1 into both equations.\u201d The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345741629\">The ordered pair (2, \u22121) made both equations true. Therefore (2, \u22121) is a solution to this system.<\/p>\n<p id=\"fs-id1168345273771\">Let\u2019s try another ordered pair. Is the ordered pair (3, 2) a solution?<\/p>\n<p><span id=\"fs-id1168345745133\" data-type=\"media\" data-alt=\"This figure begins with the sentence, \u201cWe substitute x equals 3 and y equals 2 into both equations.\u201d The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_002_img_new.jpg\" alt=\"This figure begins with the sentence, \u201cWe substitute x equals 3 and y equals 2 into both equations.\u201d The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345284173\">The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to <strong data-effect=\"bold\">both<\/strong> equations, it is not a solution to this system.<\/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 id=\"fs-id1168345276323\" data-type=\"problem\">\n<p id=\"fs-id1168345375020\">Determine whether the ordered pair is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-76519f616fc34373026c7a1e164b0f3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1168345296934\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ef9b0ead245d0b508218b1cb59d48442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ebe785a1a1f173c2cb47dc17a15cdc63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345356556\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p><span data-type=\"newline\">a)<br \/>\n<\/span><span id=\"fs-id1168345219786\" data-type=\"media\" data-alt=\"This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, \u201cWe substitute x = negative 2 and y = 1 into both equations,\u201d follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, \u201c(negative 2, negative 1) does not make both equations true.\u201d Under the second equation is the sentence, \u201c(negative 2, negative 1) is not a solution.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_003_img_new.jpg\" alt=\"This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, \u201cWe substitute x = negative 2 and y = 1 into both equations,\u201d follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, \u201c(negative 2, negative 1) does not make both equations true.\u201d Under the second equation is the sentence, \u201c(negative 2, negative 1) is not a solution.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span>(\u22122, \u22121) does not make both equations true. (\u22122, \u22121) is not a solution.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345509987\" data-type=\"media\" data-alt=\"This figure begins with the sentence, \u201cWe substitute x = -4 and y = -3 into both equations.\u201d The first equation listed shows x \u2013 y = -1. Then -4 - (-3) = -1. Then -1 = -1. The second equation listed shows 2x \u2013 y = -5. Then 2 times (-4) \u2013 (-3) = -5. Then -5 = -5. Under the first equation is the sentence, \u201c(-4, -3) does make both equations true.\u201d Under the second equation is the sentence, \u201c(-4, -3) is a solution.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_004_img_new.jpg\" alt=\"This figure begins with the sentence, \u201cWe substitute x = -4 and y = -3 into both equations.\u201d The first equation listed shows x \u2013 y = -1. Then -4 - (-3) = -1. Then -1 = -1. The second equation listed shows 2x \u2013 y = -5. Then 2 times (-4) \u2013 (-3) = -5. Then -5 = -5. Under the first equation is the sentence, \u201c(-4, -3) does make both equations true.\u201d Under the second equation is the sentence, \u201c(-4, -3) is a solution.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span>(\u22124, \u22123) does not make both equations true. (\u22124, \u22123) is a solution.<\/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 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345425152\" data-type=\"problem\">\n<p id=\"fs-id1168345436547\">Determine whether the ordered pair is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-21ed0ff845ad425d209c04741861a168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1168341916968\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-876cb3caf33e984a34d443f2b79f105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#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-id1168345441011\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345261680\"><span class=\"token\">a)<\/span> yes b) no<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Equations by Graphing<\/h1>\n<\/div>\n<div id=\"fs-id1168345440846\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345425485\">In this chapter we will use three methods to solve a system of linear equations. The first method we\u2019ll use is graphing.<\/p>\n<p id=\"fs-id1168341959249\">The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p>\n<p id=\"fs-id1168345511616\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.<\/p>\n<p id=\"fs-id1168345293314\">Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_005\">(Figure 1)<\/a>:<\/p>\n<div id=\"CNX_ElemAlg_Figure_05_01_005\" class=\"bc-figure figure\">\n<figure style=\"width: 875px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_005_img_new.jpg\" alt=\"This figure shows three x y-coordinate planes. The first plane shows two lines which intersect at one point. Under the graph it says, \u201cThe lines intersect. Intersecting lines have one point in common. There is one solution to this system.\u201d The second x y-coordinate plane shows two parallel lines. Under the graph it says, \u201cThe lines are parallel. Parallel lines have no points in common. There is no solution to this system.\u201d The third x y-coordinate plane shows one line. Under the graph it says, \u201cBoth equations give the same line. Because we have just one line, there are infinitely many solutions.\u201d\" width=\"875\" height=\"349\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1168345325810\">For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we\u2019ll use a different method in each section. After seeing the third method, you\u2019ll decide which method was the most convenient way to solve this system.<\/p>\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<div data-type=\"title\">How to Solve a System of Linear Equations by Graphing<\/div>\n<div id=\"fs-id1168345283558\" data-type=\"exercise\">\n<div id=\"fs-id1168345451825\" data-type=\"problem\">\n<p id=\"fs-id1168345250621\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-93c77aec095fb80605576f00f515b39d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345196799\" 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-id1168345644826\" data-type=\"media\" data-alt=\"This table has four rows and three columns. The first column acts as the header column. The first row reads, \u201cStep 1. Graph the first equation.\u201d Then it reads, \u201cTo graph the first line, write the equation in slope-intercept form.\u201d The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x \u2013 2y = 6 is also listed.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006a_img_new.jpg\" alt=\"This table has four rows and three columns. The first column acts as the header column. The first row reads, \u201cStep 1. Graph the first equation.\u201d Then it reads, \u201cTo graph the first line, write the equation in slope-intercept form.\u201d The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x \u2013 2y = 6 is also listed.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345191681\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2. Graph the second equation on the same rectangular coordinate system.\u201d Then it says, \u201cTo graph the second line, use intercepts.\u201d This is followed by the equation x \u2013 2y = 6 and the ordered pairs (0, -3) and (6, 0). The last column of this row shows a graph of the two equations.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2. Graph the second equation on the same rectangular coordinate system.\u201d Then it says, \u201cTo graph the second line, use intercepts.\u201d This is followed by the equation x \u2013 2y = 6 and the ordered pairs (0, -3) and (6, 0). The last column of this row shows a graph of the two equations.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345425385\" data-type=\"media\" data-alt=\"The third row reads, \u201cStep 3. Determine whether the lines intersect, are parallel, or are the same line.\u201d Then \u201cLook at the graph of the lines.\u201d Finally it reads, \u201cThe lines intersect.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006c_img_new.jpg\" alt=\"The third row reads, \u201cStep 3. Determine whether the lines intersect, are parallel, or are the same line.\u201d Then \u201cLook at the graph of the lines.\u201d Finally it reads, \u201cThe lines intersect.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168341841026\" data-type=\"media\" data-alt=\"The fourth row reads, \u201cStep 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.\u201d Then it reads, \u201cSince the lines intersect, find the point of intersection. Check the point in both equations.\u201d Finally it reads, \u201cThe lines intersect at (4, -1). It then uses substitution to show that, \u201cThe solution is (4, -1).\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_006d_img_new.jpg\" alt=\"The fourth row reads, \u201cStep 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.\u201d Then it reads, \u201cSince the lines intersect, find the point of intersection. Check the point in both equations.\u201d Finally it reads, \u201cThe lines intersect at (4, -1). It then uses substitution to show that, \u201cThe solution is (4, -1).\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\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-id1168345220500\" data-type=\"problem\">\n<p id=\"fs-id1168345431391\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-94872336f7803e39f2f581a4e139df1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345560615\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345449012\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f3b8ee106f7e5a95f1dd3e0ab5f16435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt; text-align: initial;\">The steps to use to solve a system of linear equations by graphing are shown below<\/span><\/p>\n<div id=\"fs-id1168345450126\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">To solve a system of linear equations by graphing.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169751856808\" class=\"stepwise\" type=\"1\">\n<li>Graph the first equation.<\/li>\n<li>Graph the second equation on the same rectangular coordinate system.<\/li>\n<li>Determine whether the lines intersect, are parallel, or are the same line.<\/li>\n<li>Identify the solution to the system.\n<ul id=\"fs-id1168345443090\" data-bullet-style=\"open-circle\">\n<li>If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.<\/li>\n<li>If the lines are parallel, the system has no solution.<\/li>\n<li>If the lines are the same, the system has an infinite number of solutions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345689915\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345215696\" data-type=\"exercise\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\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-id1168345689915\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345215696\" data-type=\"exercise\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\n<div id=\"fs-id1168345240470\" data-type=\"problem\">\n<p id=\"fs-id1168345450358\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4cf21a10e302f2ce44fc3ee1e369370a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#52;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345261327\" 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-id1168345360405\">Both of the equations in this system are in slope-intercept form, so we will use their slopes and <em data-effect=\"italics\">y<\/em>-intercepts to graph them. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-515225576b2a5b4bd2ea5fd14d9bbe2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#52;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167829696931\" style=\"width: 100%;\" summary=\"This figure begins with a set of instructions. The first line reads, \u201cFind the slope and y-intercept of the first equation.\u201d Next to this, it shows that y equals 2x plus 1 where m = 2 and b = 1. The next line down reads, \u201cFind the slope and y-intercept of the first equation.\u201d Next to this on the right, it shows that y = 4x \u2013 1 where m = 4 and b = -1. The next line down reads, \u201cGraph the two lines\u201d. Beneath this it reads, \u201cDetermine the point of intersection. The lines intersect at (1, 3). This it shows a graph of the two lines on an x, y coordinate plane. The lines intersect at (1, 3). Then the figure says, \u201cCheck the solution in both equations.\u201d The first equation reads, y = 2 plus 1. Then 3 = 2 times 1 plus 1. Then 3 = 3. The second equation reads, y = 4x \u2013 1. Then 3 = 4 times 1 \u2013 1. Then 3 = 3. Then it says, \u201cThe solution is (1, 3).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\"><br \/>\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836693016\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\"><br \/>\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836492140\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the two lines.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-align=\"center\" data-valign=\"top\">The lines intersect at (1, 3).<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833102398\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Check the solution in both equations.<\/td>\n<td data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75b7ffa38ef7bbff5a2ea3ab7c6f8efc_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;&#99;&#125;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&middot;&#49;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&middot;&#49;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"274\" style=\"vertical-align: -25px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\" data-valign=\"top\">The solution is (1, 3).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\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 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345193878\" data-type=\"problem\">\n<p id=\"fs-id1168345270231\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6d3b5eb85c1a6845a1a2d152d0555cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345329125\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345433895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03df60c183567a5ad79ee9c595898add_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345193878\" data-type=\"problem\"><span style=\"orphans: 1; text-align: initial; font-size: 14pt;\">Both equations in Example 3 were given in slope\u2013intercept form. This made it easy for us to quickly graph the lines. In the next example, we\u2019ll first re-write the equations into slope\u2013intercept form.<\/span><\/div>\n<\/div>\n<\/div>\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-id1168345301935\" data-type=\"problem\">\n<p id=\"fs-id1168345250708\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f00ab61c2392cd4cd04c9e2e92c6756a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345302868\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167836326706\">We\u2019ll solve both of these equations for <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;\" \/> so that we can easily graph them using their slopes and <em data-effect=\"italics\">y<\/em>-intercepts. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95a3f02a781e4885b83febfed2ac6d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167824584625\" style=\"width: 100%;\" summary=\"This figure shows a solution in two columns. The first line in the left column, reads \u201cSolve the first equation for y.\u201d To the right of this, it has the equation 3x + y = negative 1 which becomes y = negative 3x \u2013 1. The next line down reads, \u201cFind the slope and y-intercept.\u201d Then it shows that m = negative 3 and b = negative 1. Below this reads, \u201cSolve the second equation for y.\u201d Next to this is the equation 2x + 1 = 0 become y = negative 2x. Below this reads, \u201cFind the slope and y-intercept.\u201d Next to this it shows that b = negative 2 and b = 0. Then it says, \u201cGraph the lines.\u201d An x y-coordinate plane shows the two graphed lines intersecting at (negative 1, 2). The figure then indicates, \u201cDetermine the point of intersection. The lines intersect at (negative 1, 2).\u201d Finally the figure says, \u201cCheck the solution in both equations.\u201d The first equation shows 3x + y = -1. Then 3(-1) + 2 = -1. And then -1 = -1. The second equation shows 2x + y = 0. Then 2(-1) + 2 = 0. Then 0 = 0. The figure then says, \u201cThe solution is (-1, 2).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Solve the second equation for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1fb0aa089df7ad1b7f98426d2862a83d_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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"254\" width=\"159\" style=\"vertical-align: -110px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836705596\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-align=\"center\" data-valign=\"top\">The lines intersect at (\u22121, 2).<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Check the solution in both equations.<\/td>\n<td data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ebed3ab2e033d31726b97600aaf2a2ac_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;&#99;&#125;&#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;&#51;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"350\" style=\"vertical-align: -26px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\" data-valign=\"top\">The solution is (\u22121, 2).<\/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 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345337170\" data-type=\"problem\">\n<p id=\"fs-id1168345301261\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2224b844096f02f3dc2d954ec08eb05d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#120;&#43;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345561188\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345254462\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aecc845ff8fcecb422091e6436adca8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345398741\">Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We\u2019ll do this in the next example.<\/p>\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-id1168345644007\" data-type=\"problem\">\n<p id=\"fs-id1168345695461\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0adcda6dc53d147fe9d2e9937765bcee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"105\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345273530\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1168345276953\">We will find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts of both equations and use them to graph the lines.<\/p>\n<table id=\"fs-id1167829578991\" style=\"width: 100%;\" summary=\"This is a figure that shows a solution in three columns. In the first column, it reads, \u201cTo find the intercepts, let x = 0 and solve for y, then let y = 0 and solve for x.\u201d The middle column shows the equation x + y = 2. Centered below this it reads x + y = 2 and next to this, x + y = 2. Below this it shows 0 + y = 2, and y = 2. Then it shows x + 0 = 2, and x = 2. There is a table to the right with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the number 0 and 2. Under \u201cy\u201d are the numbers 2 and 0. In the next row down, it reads, \u201cFind the intercepts, let x = 0 then let y = 0.\u201d The middle column shows x - y = 4. Then x - y = 4 and x - y = 4. Below this it shows 0 - y = 4, -y = 4, and y = negative 4. Then it shows x - 0 = 4, and x = 4. There is a table to the right with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the number 0 and 4. Under \u201cy\u201d are the numbers -4 and 0. Below this there is a graph that shows two lines intersecting at point (3, -1) on an x y-coordinate plane.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836629538\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">To find the intercepts, let <em data-effect=\"italics\">x<\/em> = 0 and solve<span data-type=\"newline\"><br \/>\n<\/span>for <em data-effect=\"italics\">y<\/em>, then let <em data-effect=\"italics\">y<\/em> = 0 and solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6228fb232396dcb697184a08bb055bb8_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;&#99;&#125;&#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;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#48;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"256\" style=\"vertical-align: -26px;\" \/><\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><span id=\"fs-id1167829596067\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836621931\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">To find the intercepts, let<span data-type=\"newline\"><br \/>\n<\/span><em data-effect=\"italics\">x<\/em> = 0 then let <em data-effect=\"italics\">y<\/em> = 0.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cf43e732d1ed1a745309c9ec8187f394_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;&#99;&#125;&#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;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"271\" style=\"vertical-align: -37px;\" \/><\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829830534\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_010b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Graph the line.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><span id=\"fs-id1167836320119\" data-type=\"media\" data-alt=\"This graph shows two lines intersection at point (3, -1) on an x y-coordinate plane.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_011_img_new.jpg\" alt=\"This graph shows two lines intersection at point (3, -1) on an x y-coordinate plane.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 19.2046%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Determine the point of intersection.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\">The lines intersect at (3, \u22121).<\/td>\n<td style=\"width: 19.2046%;\" data-valign=\"top\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.9091%;\" data-valign=\"top\">Check the solution in both equations.<\/td>\n<td style=\"width: 64.7727%;\" data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-11f18a385c0b8927899faf46f13fd43e_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;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"293\" style=\"vertical-align: -26px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>The solution is (3, \u22121).<\/td>\n<td style=\"width: 19.2046%;\"><\/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 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345727147\" data-type=\"problem\">\n<p id=\"fs-id1168345301947\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c07cfb0dcc6d523f7383696050760ffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"105\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345292167\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345390050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eb6560ec04491f4e7cf02e14e6df5ec3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345389141\">Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.<\/p>\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-id1168345431017\" data-type=\"problem\">\n<p id=\"fs-id1168345419137\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8fa058c2e85f15c16790fc72fbd1a00a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345287687\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836356457\" style=\"width: 100%;\" summary=\"This figure shows two equations: y = 6 and x plus 3y = 12. It then says, \u201cWe know the first equation represents a horizontal line whose y-intercept is 6.\u201d Then is shows that y = 6. It then says, \u201cThe second equation is most conveniently graphed using intercepts.\u201d Then it shows 2x plus 3y = 12. Then it reads, \u201cTo find the intercepts, let x = 0 and then y = 0.\u201d There is a graph with the columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and 6. Under \u201cy\u201d are the numbers 4 and 0. Then it reads, \u201cGraph the lines.\u201d The two lines are graphed on an x y-coordinate plane. The lines intersect at point (negative 3, 6). The figure then reads, \u201cDetermine the point of intersection. The lines intersect at (negative 3, 6).\u201d The figure then shows that y = 6, then 6 = 6, and 2 = 2. It also shows that 2x +3y = 12. Then, 2 times negative 3, in parentheses, plus 3 times 6 = 12. Then negative 6 plus 18 = 12, and 12 = 12. The figure then indicates, \u201cThe solution is (negative 3, 6).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 24.2045%;\"><\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167829594693\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">We know the first equation represents a horizontal<span data-type=\"newline\"><br \/>\n<\/span>line whose <em data-effect=\"italics\">y<\/em>-intercept is 6.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836550964\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">The second equation is most conveniently graphed<span data-type=\"newline\"><br \/>\n<\/span>using intercepts.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836492417\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">To find the intercepts, let <em data-effect=\"italics\">x<\/em> = 0 and then <em data-effect=\"italics\">y<\/em> = 0.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167836355778\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Graph the lines.<\/td>\n<td style=\"width: 75.6818%;\" data-valign=\"top\"><span id=\"fs-id1167829790109\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_012e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Determine the point of intersection.<\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\">The lines intersect at (\u22123, 6).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\" data-valign=\"top\">Check the solution to both equations.<\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0448b6d152f331f059b22cd0f33e65d7_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;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#51;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#43;&#49;&#56;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"281\" style=\"vertical-align: -39px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.2045%;\"><\/td>\n<td style=\"width: 75.6818%;\" data-align=\"center\" data-valign=\"top\">The solution is (\u22123, 6).<\/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 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345250431\" data-type=\"problem\">\n<p id=\"fs-id1168345230324\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b2c379455bcde2d5ac1e3daaf6d8342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345229034\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345286142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0ef2cb721f4e314e492442d7e4973839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt; text-align: initial;\">In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we\u2019ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.<\/span><\/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 id=\"fs-id1168341955833\" data-type=\"problem\">\n<p id=\"fs-id1168341864012\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea4118511a2c356ccdff5a6a4b720570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345260776\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836599119\" style=\"width: 100%;\" summary=\"This figure lists two equations: y = one half \u201cx\u201d minus 3 and x minus 2 = 4. It then indicates, \u201cTo graph the first equation, we will use the slope and y-intercept.\u201d The figure then shows that y = one half \u201cx\u201d minus 3 in which m = one half and b = negative 3. The next line down reads, \u201cTo graph the second equation, we will use the intercepts.\u201d Next to this is the equation x minus 2y = 4 which is followed by a table with two columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and 4. Under \u201cy\u201d are the numbers negative 2 and 0. The figure then says, \u201cGraph the lines.\u201d The two lines are then graphed on an x y-coordinate plane. The two lines appear to be parallel. The figure then reads, \u201cDetermine the point of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167825829989\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">To graph the first equation, we will<span data-type=\"newline\"><br \/>\n<\/span>use its slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829877876\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829753121\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836557401\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">To graph the second equation,<span data-type=\"newline\"><br \/>\n<\/span>we will use the intercepts.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836543355\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833022880\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836366683\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_013g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-valign=\"top\">\u2003\u2003\u2003\u2003The lines are parallel.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Since no point is on both lines, there is no ordered pair<span data-type=\"newline\"><br \/>\n<\/span>that makes both equations true. There is no solution to<span data-type=\"newline\"><br \/>\n<\/span>this system.<\/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 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345746509\" data-type=\"problem\">\n<p id=\"fs-id1168345420434\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-93464f18895fb875b7f5621896581607_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#52;&#121;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345725138\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345418010\">no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916918\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168341913823\" data-type=\"exercise\">\n<div id=\"fs-id1168345746509\" data-type=\"problem\">\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-id1168345194746\" data-type=\"problem\">\n<p id=\"fs-id1168345428978\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-007b48e429a6e87df1723b5dc5ca1841_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#54;&#120;&#43;&#51;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"150\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345510019\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836510916\" style=\"width: 100%;\" summary=\"This figure shows two equations: y = 2x minus 3 and negative 6x plus 3y = negative 9. The next line down reads, \u201cFind the slope and y-intercept of the first equation.\u201d It then shows that y = 2x -3 where m = 2 and b = negative 3. I also shows negative 6x plus 3y = negative 9. It then reads, \u201cFind the intercepts of the second equation.\u201d There is a table with two columns labeled \u201cx\u201d and \u201cy.\u201d Under \u201cx\u201d are the numbers 0 and three over two. Under \u201cy\u201d are the numbers negative 3 and 0. The figure then reads, \u201cGraph the lines.\u201d There appears to be only one line graphed on the x y coordinate plane. The figure then reads, \u201cDetermine the point of intersection. The lines are the same! Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to this system.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833380846\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the<span data-type=\"newline\"><br \/>\n<\/span>first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836628539\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Find the intercepts of the second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829599268\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833139693\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph the lines.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836287817\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_014e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Determine the point of intersection.<\/td>\n<td data-valign=\"top\">The lines are the same!<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Since every point on the line makes both equations<span data-type=\"newline\"><br \/>\n<\/span>true, there are infinitely many ordered pairs that make<span data-type=\"newline\"><br \/>\n<\/span>both equations true.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">There are infinitely many solutions to this system.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345414817\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345650454\" data-type=\"exercise\">\n<div id=\"fs-id1168345302969\" data-type=\"solution\">\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-id1168345303045\" data-type=\"problem\">\n<p id=\"fs-id1168345445689\">Solve each system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3d13a2aff02eb9c7d37e43da6e79158c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#51;&#120;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#50;&#121;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"145\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345407885\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345453131\">infinitely many solutions<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt; text-align: initial;\">If you write the second equation in Example 8 in slope-intercept form, you may recognize that the equations have the same slope and same <\/span><em style=\"font-size: 14pt; text-align: initial;\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt; text-align: initial;\">-intercept.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345329224\">When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Coincident Lines<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Coincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341840730\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Determine the Number of Solutions of a Linear System<\/h1>\n<p id=\"fs-id1168345262002\">There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.<\/p>\n<p id=\"fs-id1168345325732\">We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example 2 through Example 6 all had two intersecting lines. Each system had one solution.<\/p>\n<p id=\"fs-id1168345230044\">A system with parallel lines, like Example 7, has no solution. What happened in Example 8? The equations have <span class=\"no-emphasis\" data-type=\"term\">coincident lines<\/span>, and so the system had infinitely many solutions.<\/p>\n<p id=\"fs-id1168345237685\">We\u2019ll organize these results in <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_018_img\">(Table 1)<\/a> below:<\/p>\n<div id=\"CNX_ElemAlg_Figure_05_01_018_img\" class=\"bc-figure figure\">\n<figure style=\"width: 317px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_018_img_new.jpg\" alt=\"This table has two columns and four rows. The first row labels each column \u201cGraph\u201d and \u201cNumber of solutions.\u201d Under \u201cGraph\u201d are \u201c2 intersecting lines,\u201d \u201cParallel lines,\u201d and \u201cSame line.\u201d Under \u201cNumber of solutions\u201d are \u201c1,\u201d \u201cNone,\u201d and \u201cInfinitely many.\u201d\" width=\"317\" height=\"121\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Table 1<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1168345693726\">Parallel lines have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. So, if we write both equations in a system of linear equations in slope\u2013intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 7.<\/p>\n<div id=\"fs-id1168345743284\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bd2fceef87fc954c468c9416bd236648_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;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#108;&#105;&#110;&#101;&#32;&#105;&#115;&#32;&#105;&#110;&#32;&#115;&#108;&#111;&#112;&#101;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#32;&#119;&#101;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#101;&#32;&#103;&#101;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#98;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#98;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"154\" width=\"696\" style=\"vertical-align: -72px;\" \/><\/div>\n<p id=\"fs-id1168345287565\">The two lines have the same slope but different <em data-effect=\"italics\">y<\/em>-intercepts. They are parallel lines.<\/p>\n<p id=\"fs-id1168345453187\"><a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_019_img\">(Table 2)<\/a> shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.<\/p>\n<div id=\"CNX_ElemAlg_Figure_05_01_019_img\" class=\"bc-figure figure\">\n<figure style=\"width: 486px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_019_img_new.jpg\" alt=\"This table is entitled \u201cNumber of Solutions of a Linear System of Equations.\u201d There are four columns. The columns are labeled, \u201cSlopes,\u201d \u201cIntercepts,\u201d \u201cType of Lines,\u201d \u201cNumber of Solutions.\u201d Under \u201cSlopes\u201d are \u201cDifferent,\u201d \u201cSame,\u201d and \u201cSame.\u201d Under \u201cIntercepts,\u201d the first cell is blank, then the words \u201cDifferent\u201d and \u201cSame\u201d appear. Under \u201cTypes of Lines\u201d are the words, \u201cIntersecting,\u201d \u201cParallel,\u201d and \u201cCoincident.\u201d Under \u201cNumber of Solutions\u201d are \u201c1 point,\u201d \u201cNo Solution,\u201d and \u201cInfinitely many solutions.\u201d\" width=\"486\" height=\"148\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Table 2<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1168345287740\">Let\u2019s take one more look at our equations in <a class=\"autogenerated-content\" href=\"#fs-id1168341955833\">(Example 7)<\/a> that gave us parallel lines.<\/p>\n<div id=\"fs-id1168345510136\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6156bc2e1c692f7efa35a64e1bd0b9e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"104\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168345650553\">When both lines were in slope-intercept form we had:<\/p>\n<div id=\"fs-id1168345302936\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-932acf02431f78a427eaadb1633b0302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"203\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1168345457988\">Do you recognize that it is impossible to have a single 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;\" \/> that is a solution to both of those equations?<\/p>\n<p id=\"fs-id1168345287296\">We call a system of equations like this <span class=\"no-emphasis\" data-type=\"term\">an inconsistent system<\/span>. It has no solution.<\/p>\n<p id=\"fs-id1168345692186\">A system of equations that has at least one solution is called a <span class=\"no-emphasis\" data-type=\"term\">consistent system<\/span>.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Consistent and Inconsistent Systems<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1168345692954\">A <span data-type=\"term\">consistent system<\/span> of equations is a system of equations with at least one solution.<\/p>\n<p id=\"fs-id1168345427031\">An <span data-type=\"term\">inconsistent system<\/span> of equations is a system of equations with no solution.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345240009\">We also categorize the equations in a system of equations by calling the equations <em data-effect=\"italics\">independent<\/em> or <em data-effect=\"italics\">dependent<\/em>. If two equations are <span data-type=\"term\">independent equations<\/span>, they each have their own set of solutions. Intersecting lines and parallel lines are independent.<\/p>\n<p id=\"fs-id1168345228631\">If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two <span class=\"no-emphasis\" data-type=\"term\">dependent equations<\/span>, we get coincident lines.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Independent and Dependent Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1168345423433\">Two equations are <span data-type=\"term\">independent<\/span> if they have different solutions.<\/p>\n<p id=\"fs-id1168345278054\">Two equations are <span data-type=\"term\">dependent<\/span> if all the solutions of one equation are also solutions of the other equation.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345417086\">Let\u2019s sum this up by looking at the graphs of the three types of systems. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_015\">(Figure 3)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_05_01_020_img\">(Table 3)<\/a>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_05_01_015\" class=\"bc-figure figure\">\n<figure style=\"width: 791px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_015_img_new.jpg\" alt=\"This figure shows three x y coordinate planes in a horizontal row. The first shows two lines intersecting. The second shows two parallel lines. The third shows two coincident lines.\" width=\"791\" height=\"257\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_05_01_020_img\" class=\"bc-figure figure\"><\/div>\n<div class=\"bc-figure figure\">\n<figure style=\"width: 528px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_020_img_new.jpg\" alt=\"This table has four columns and four rows. The columns are labeled, \u201cLines,\u201d \u201cIntersecting,\u201d \u201cParallel,\u201d and \u201cCoincident.\u201d In the first row under the labeled column \u201clines\u201d it reads \u201cNumber of solutions.\u201d Reading across, it tell us that an intersecting line contains 1 point, a parallel line provides no solution, and a coincident line has infinitely many solutions. A consistent\/inconsistent line has consistent lines if they are intersecting, inconsistent lines if they are parallel and consistent if the lines are coincident. Finally, dependent and independent lines are considered independent if the lines intersect, they are also independent if the lines are parallel, and they are dependent if the lines are coincident.\" width=\"528\" height=\"122\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Table 3<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div>\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-id1168345448163\" data-type=\"problem\">\n<p id=\"fs-id1168345671375\">Without graphing, determine the number of solutions and then classify the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-577f8d3ab3cc62dbd605b75bef663c52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#50;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345251139\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-118\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We will compare the slopes and intercepts of the two lines.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-577f8d3ab3cc62dbd605b75bef663c52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#50;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The first equation is already in slope-intercept form.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-09acfc77aa1f355947c21a5c0e345588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the second equation in slope-intercept form.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0f1ebb7cf4fa60764b7738fdc4143e90_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;&#54;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#120;&#43;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#121;&#125;&#123;&#45;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#120;&#43;&#49;&#50;&#125;&#123;&#45;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"177\" style=\"vertical-align: -38px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the slope and intercept of each line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3c92577b0939cac2035a0bfda47650d5_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;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#54;&#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;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"288\" style=\"vertical-align: -23px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>Since the slopes are the same and <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;\" \/>-intercepts are different, the lines are parallel.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345560002\">A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.<\/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-id1168345326624\" data-type=\"problem\">\n<p id=\"fs-id1168345416501\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345224525\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-268ac950028533cb2a6f7e84f4c6d3f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#50;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#50;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345415558\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345667907\">no solution, inconsistent, independent<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341840730\" class=\"bc-section section\" data-depth=\"1\">\n<div id=\"fs-id1168345538684\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345408145\" data-type=\"exercise\">\n<div id=\"fs-id1168345414201\" data-type=\"solution\">\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-id1168341857668\" data-type=\"problem\">\n<p id=\"fs-id1168341857670\">Without graphing, determine the number of solutions and then classify the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-baac1a1f024d68ba74ba4c8454092b38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#53;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345553490\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-504\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We will compare the slope and intercepts of the two lines.<\/td>\n<td colspan=\"2\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a14d11691ed831db79ccb667551d56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#53;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write both equations in slope-intercept form.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4740b37c6d9604ced5b71c1aaeb5d177_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;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#38;&#32;&#45;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"160\" style=\"vertical-align: -15px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7eb39cf851135e44a20ada683b168bec_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;&#120;&#45;&#53;&#121;&#61;&#53;&#38;&#32;&#61;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#121;&#38;&#32;&#61;&#38;&#32;&#45;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#53;&#121;&#125;&#123;&#45;&#53;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#120;&#43;&#53;&#125;&#123;&#45;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"182\" style=\"vertical-align: -40px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the slope and intercept of each line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-270830f55dd1da90c55b95646b8e51a3_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;&#121;&#38;&#32;&#61;&#38;&#32;&#45;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"125\" style=\"vertical-align: -22px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5b8586f69d16fb4ed78c76f8769f7fd2_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;&#121;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#38;&#32;&#45;&#49;&#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=\"112\" style=\"vertical-align: -23px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td colspan=\"2\">Since the slopes are different, the lines intersect.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345742349\">A system of equations whose graphs are intersect has 1 solution and is consistent and independent.<\/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 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345417929\" data-type=\"problem\">\n<p id=\"fs-id1168345398213\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345398216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d78dd6d3e1b54ea94cf3eaff925b2bac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#50;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345458749\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345458751\">one solution, consistent, independent<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345425501\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345417927\" data-type=\"exercise\">\n<div id=\"fs-id1168345417929\" data-type=\"problem\">\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<div id=\"fs-id1168345557327\" data-type=\"problem\">\n<p id=\"fs-id1168345557329\">Without graphing, determine the number of solutions and then classify the system of equations. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a36f11fc176a9c2a264aa6fca7651e94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345384539\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-238\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We will compare the slopes and intercepts of the two lines.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a36f11fc176a9c2a264aa6fca7651e94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the first equation in slope-intercept form.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-59dfa126495322b6cd1daf50f4e49c5b_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;&#51;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#121;&#125;&#123;&#45;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#120;&#43;&#52;&#125;&#123;&#45;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"169\" style=\"vertical-align: -40px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The second equation is already in slope-intercept form.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ab41e0e98df3824af4f9c90ef62a0cf1_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;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>Since the slopes are the same, they have the same slope and same <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;\" \/>-intercept and so the lines are coincident.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168341852918\">A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345458749\" data-type=\"solution\">\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-id1168345374488\" data-type=\"problem\">\n<p id=\"fs-id1168345374490\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1168345398366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-31501a5b039c47fd24ba363656ebfc95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#53;&#121;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345428781\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345428783\">infinitely many solutions, consistent, dependent<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916335\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Graphing<\/h1>\n<p id=\"fs-id1168345743034\">We will modify the\u00a0 problem solving strategy slightly to set up and solve applications of systems of linear equations.<\/p>\n<div id=\"fs-id1168341972806\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How to use a problem solving strategy for systems of linear equations.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341972806\" class=\"howto\" data-type=\"note\">\n<ol id=\"fs-id1169751898179\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1168341910497\">Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345415363\" data-type=\"problem\">\n<p id=\"fs-id1168345415365\">Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?<\/p>\n<\/div>\n<div id=\"fs-id1168345667515\" 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-id1168341961495\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/p>\n<p id=\"fs-id1168341841009\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/p>\n<p id=\"fs-id1168345447831\">We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.<\/p>\n<p id=\"fs-id1168345447835\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/p>\n<p id=\"fs-id1168037092275\">\u2003\u2003Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e357bc804ed493fa8caf9a607e52703d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"29\" style=\"vertical-align: -4px;\" \/> number of quarts of fruit juice.<span data-type=\"newline\"><br \/>\n<\/span> \u2003\u2003\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3ccd20695087873d8035662126b0a7a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> number of quarts of club soda<\/p>\n<p id=\"fs-id1168341958682\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span id=\"fs-id1168341917919\" data-type=\"media\" data-alt=\"This figure shows sentences converted into equations. The first sentence reads, \u201cThe number of quarts of fruit juice and the number of quarts of club soda is 10. \u201cNumber of quarts of fruit juice\u201d contains a curly bracket beneath the phrase with an \u201cf\u201d centered under the bracket. The \u201cAnd\u201d also contains a curly bracket beneath it and has a plus sign centered beneath it. \u201cNumber of quarts of club soda\u201d contains a curly bracket with the variable \u201cc\u201d beneath it. And finally, the phrase \u201cis 10\u201d contains a curly bracket. Under this it reads equals 10. The second sentence reads, \u201cThe number of quarts of fruit juice is four times the number of quarts of club soda\u201d. This sentence is set up similarly in that each phrase contains a curly bracket underneath. The variable \u201cf\u201d represents \u201cThe number of quarts of fruit juice\u201d. An equal sign represents \u201cis\u201d and \u201c4c\u201d represents four times the number of quarts of club soda.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_016_img_new.jpg\" alt=\"This figure shows sentences converted into equations. The first sentence reads, \u201cThe number of quarts of fruit juice and the number of quarts of club soda is 10. \u201cNumber of quarts of fruit juice\u201d contains a curly bracket beneath the phrase with an \u201cf\u201d centered under the bracket. The \u201cAnd\u201d also contains a curly bracket beneath it and has a plus sign centered beneath it. \u201cNumber of quarts of club soda\u201d contains a curly bracket with the variable \u201cc\u201d beneath it. And finally, the phrase \u201cis 10\u201d contains a curly bracket. Under this it reads equals 10. The second sentence reads, \u201cThe number of quarts of fruit juice is four times the number of quarts of club soda\u201d. This sentence is set up similarly in that each phrase contains a curly bracket underneath. The variable \u201cf\u201d represents \u201cThe number of quarts of fruit juice\u201d. An equal sign represents \u201cis\u201d and \u201c4c\u201d represents four times the number of quarts of club soda.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345374822\">We now have the system. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-455195821f88a4f9a998b5abedb999c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#102;&#43;&#99;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#61;&#52;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1168345558568\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations using good algebra techniques.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span id=\"fs-id1168345634396\" data-type=\"media\" data-alt=\"This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_017_img_new.jpg\" alt=\"This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345386283\">The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.<\/p>\n<p id=\"fs-id1168345329449\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make sure it makes sense.<\/p>\n<p id=\"fs-id1168345342682\">Does this make sense in the problem?<\/p>\n<p id=\"fs-id1168345342685\">Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.<\/p>\n<p id=\"fs-id1168345457793\">Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.<\/p>\n<p id=\"fs-id1168345457796\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question with a complete sentence.<\/p>\n<p id=\"fs-id1168345425413\">Sondra needs 8 quarts of fruit juice and 2 quarts of soda.<\/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 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345461354\" data-type=\"problem\">\n<p id=\"fs-id1168345461356\">Manu is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manu need?<\/p>\n<\/div>\n<div id=\"fs-id1168345748496\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345748498\">Manu needs 3 quarts juice concentrate and 9 quarts water.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341952013\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1168345745168\">Access these online resources for additional instruction and practice with solving systems of equations by graphing.<\/p>\n<ul id=\"fs-id1168345745172\" data-display=\"block\">\n<li><a href=\"http:\/\/www.openstax.org\/l\/25linsysGraph\">Instructional Video Solving Linear Systems by Graphing<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25solvesbyGraph\">Instructional Video Solve by Graphing<\/a><\/li>\n<\/ul>\n<h1>Key Concepts<\/h1>\n<ul>\n<li><strong data-effect=\"bold\">Problem Solving Strategy for Systems of Linear Equations<\/strong>\n<ol id=\"fs-id1169754134591\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1 data-type=\"glossary-title\">Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1168345631530\">\n<dt>coincident lines<\/dt>\n<dd id=\"fs-id1168345527779\">Coincident lines are lines that have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345527788\">\n<dt>consistent system<\/dt>\n<dd id=\"fs-id1168345527794\">A consistent system of equations is a system of equations with at least one solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345527798\">\n<dt>dependent equations<\/dt>\n<dd id=\"fs-id1168345484131\">Two equations are dependent if all the solutions of one equation are also solutions of the other equation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484135\">\n<dt>inconsistent system<\/dt>\n<dd id=\"fs-id1168345484140\">An inconsistent system of equations is a system of equations with no solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484145\">\n<dt>independent equations<\/dt>\n<dd id=\"fs-id1168345484150\">Two equations are independent if they have different solutions.<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345484154\">\n<dt>solutions of a system of equations<\/dt>\n<dd id=\"fs-id1168345693483\">Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (<em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>).<\/dd>\n<\/dl>\n<dl id=\"fs-id1168345693499\">\n<dt>system of linear equations<\/dt>\n<dd id=\"fs-id1168345363996\">When two or more linear equations are grouped together, they form a system of linear equations.<\/dd>\n<\/dl>\n<\/div>\n<h1 data-type=\"glossary-title\"><span style=\"font-size: 1.2em; word-spacing: normal;\">4.1 Exercise Set<\/span><\/h1>\n<div id=\"fs-id1168345433887\" class=\"practice-perfect\" data-depth=\"2\">\n<p id=\"fs-id1168345500508\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d854d19255e3c6bf6fd214d7e3985a4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#54;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#52;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/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;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b7ed23914456620d1ad9217ebea92dbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5188a3cfb215bb77f17514e0b146595f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-200c08ae9f5b63c7032ea221a5e44b10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"92\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-edea40c117a1cc66603e136200800f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b41e214fd7bce792b1d8d237b99971a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c5df3c05e7711ce1203981eeb9d13097_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#53;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5cb851035e4c797f673a747f65991990_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>(\\left(\\frac{5}{4},\\frac{7}{4}\\right)\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1168345452960\">In the following exercises, solve the following systems of equations by graphing.<\/p>\n<ol class=\"twocolumn\" start=\"5\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-efc0cfb8676e8a73d602912943d3e935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ed92239b065a1ff0d22dfcadfaa3892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-083748f580c44ff95b70afa809fc6fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#50;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc4bd1c4ae2451fb67dd2067b7e61545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a9b5c40555a3b204460c01bc5b89ecf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#52;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"121\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ed92239b065a1ff0d22dfcadfaa3892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eeb6af34321eaa082869ae6a63b9a3cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f146b3608b0699ba40eef3769efbcaa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-04a56c708c56f277b40448fa06ee53e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ef8b059c8d711ea0a24f39236e85586b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-045e9a4b173ce0a25e47612b3402a1b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7e7dfb7f735c5f3130a11c8491c8b95e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a0d3ff603913a65c90e46fef96567ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4380a818ca37533243434f19738bf82e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d48adacd58e148299ab3800dc5ecabc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a3fb3edcc35e3de260d8c65b58c4563e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#56;&#120;&#45;&#54;&#121;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ae045e47076c6ed872912536093bb17f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#120;&#43;&#52;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a432602acf19e64ccd5de9199215d03a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#45;&#51;&#121;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#54;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-df90e25f730a4be99a1e5e4b138b1182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#56;&#120;&#45;&#52;&#121;&#61;&#45;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"148\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3ff7e7386da699d454eda078951d424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"127\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9a8d07adcb7d479ce46952cc541623aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#50;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"138\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<div id=\"fs-id1168345521355\" data-type=\"exercise\">\n<div id=\"fs-id1168345521357\" data-type=\"problem\">\n<p id=\"fs-id1168345521359\"><span style=\"text-align: initial; font-size: 14pt;\">Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.<\/span><\/p>\n<ol class=\"twocolumn\" start=\"26\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-537cc7431b0791faa5d5e0d218532558_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ec6bfdc1ceba76f53027c5160cbe48bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#50;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#50;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/li>\n<li>missing<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-47f8388a9b26d25d17ddd768e299f102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#50;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9b10f580d6d4fb5e32fad3e0cd050d5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf126e5fd78a2347768d4a523f12b2a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345557294\" data-type=\"exercise\">\n<div id=\"fs-id1168345557296\" data-type=\"problem\">\n<p id=\"fs-id1168345557298\"><span style=\"text-align: initial; font-size: 14pt;\">In the following exercises, solve.<\/span><\/p>\n<ol start=\"32\">\n<li>Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?<\/li>\n<li>Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?<\/li>\n<li>Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345572431\" class=\"writing\" data-depth=\"2\">\n<div id=\"fs-id1168341864103\" data-type=\"exercise\">\n<div id=\"fs-id1168341864105\" data-type=\"problem\">\n<h1>Answers<\/h1>\n<ol class=\"twocolumn\">\n<li><span style=\"font-family: inherit; font-size: inherit; background-color: initial;\">a) yes b) <\/span><span style=\"font-family: inherit; font-size: inherit; background-color: initial;\">no<\/span><\/li>\n<li><span class=\"token\">a)<\/span> yes b) no<\/li>\n<li><span class=\"token\">a)<\/span> yes b) no<\/li>\n<li><span class=\"token\">a)<\/span> no b) yes<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6fd7f677a681964debbd5fb9bbb3c944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><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;\" \/><\/li>\n<li><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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-25c4864d7eaa7ae5b2fe81ae29cf46af_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f3b8ee106f7e5a95f1dd3e0ab5f16435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-745bd1150b5c3e65ae8bad5282a5b3b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/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;\" \/><\/li>\n<li><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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee47278eba3e60a9253d35f5859bb2e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e228594fa0479f071602af78e20058a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f3b8ee106f7e5a95f1dd3e0ab5f16435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-450ee1faefebebc715b20a97daae94ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e0819cfb987fa92d333add15bb2e864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>no solution<\/li>\n<li>no solution<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-df90e25f730a4be99a1e5e4b138b1182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#56;&#120;&#45;&#52;&#121;&#61;&#45;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"148\" style=\"vertical-align: -17px;\" \/><\/li>\n<li>infinitely many solutions<\/li>\n<li>infinitely many solutions<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f78f604644c2cdddbea2fc4d8ad49cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>no solutions<\/li>\n<li>no solutions<\/li>\n<li>no solutions, inconsistent, independent<\/li>\n<li>consistent, 1 solution<\/li>\n<li>infinitely many solutions<\/li>\n<li>infinitely many solutions<\/li>\n<li>Molly needs 16 ounces of strawberry juice and 48 ounces of water.<\/li>\n<li>Enrique needs 8 ounces of nuts and 16 ounces of water.<\/li>\n<li>Leo should plant 50 tulips and 300 daffodils.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":125,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["lynn-marecek","maryanne-anthony-smith"],"pb_section_license":""},"chapter-type":[],"contributor":[63,64],"license":[],"class_list":["post-1136","chapter","type-chapter","status-publish","hentry","contributor-lynn-marecek","contributor-maryanne-anthony-smith"],"part":1094,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1136","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":1,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1136\/revisions"}],"predecessor-version":[{"id":1137,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1136\/revisions\/1137"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/1094"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1136\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1136"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1136"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1136"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}