{"id":1244,"date":"2019-07-29T23:24:45","date_gmt":"2019-07-29T23:24:45","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-substitution\/"},"modified":"2021-08-31T21:20:46","modified_gmt":"2021-08-31T21:20:46","slug":"solve-systems-of-equations-by-substitution","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-substitution\/","title":{"raw":"4.2  Solve Systems of Equations by Substitution","rendered":"4.2  Solve Systems of Equations by Substitution"},"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>Solve a system of equations by substitution<\/li>\n \t<li>Solve applications of systems of equations by substitution<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345194538\">Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> both between \u221210 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.<\/p>\n<p id=\"fs-id1168345297967\">In this section, we will solve systems of linear equations by the substitution method.<\/p>\n\n<div id=\"fs-id1168345216137\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Equations by Substitution<\/h1>\n<p id=\"fs-id1168345301534\">We will use the same system we used first for graphing.<\/p>\n\n<div id=\"fs-id1168345415079\" 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-id1168345249949\">We will first solve one of the equations for either <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>. We can choose either equation and solve for either variable\u2014but we\u2019ll try to make a choice that will keep the work easy.<\/p>\n<p id=\"fs-id1168345644836\">Then we substitute that expression into the other equation. The result is an equation with just one variable\u2014and we know how to solve those!<\/p>\n<p id=\"fs-id1168345416782\">After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.<\/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 data-type=\"title\">How to Solve a System of Equations by Substitution<\/div>\n<div id=\"fs-id1168341959560\" data-type=\"exercise\">\n<div id=\"fs-id1168345656724\" data-type=\"problem\">\n<p id=\"fs-id1168345724784\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1171792480266\"><\/p>\n\n<\/div>\n<div id=\"fs-id1168345240310\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<span id=\"fs-id1168345291516\" data-type=\"media\" data-alt=\"This figure has three columns and six rows. The first row says, \u201cStep 1. Solve one of the equations for either variable.\u201d To the right of this, the middl row reads, \u201cWe\u2019ll solve the first equation for y.\u201d The third column shows the two equations: 2x + y = 7 and x \u2013 2y = 6. It shows that 2x + y = 7 becomes y = 7 \u2013 2x.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_02_001a_img_new.jpg\" alt=\"This figure has three columns and six rows. The first row says, \u201cStep 1. Solve one of the equations for either variable.\u201d To the right of this, the middl row reads, \u201cWe\u2019ll solve the first equation for y.\u201d The third column shows the two equations: 2x + y = 7 and x \u2013 2y = 6. It shows that 2x + y = 7 becomes y = 7 \u2013 2x.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345646781\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2. Substitute the expression from Step 1 into the other equation.\u201d Then, \u201cWe replace y in the second equation with the expression 7 \u2013 2x.\u201d It then shows the x \u2013 2y = 6 becomes x \u2013 2(7 \u2013 2x) = 6.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2. Substitute the expression from Step 1 into the other equation.\u201d Then, \u201cWe replace y in the second equation with the expression 7 \u2013 2x.\u201d It then shows the x \u2013 2y = 6 becomes x \u2013 2(7 \u2013 2x) = 6.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345291590\" data-type=\"media\" data-alt=\"The third row says, \u201cStep 3: Solve the resulting equation.\u201d Then \u201cNow we have an equation with just 1 variable. WE know how to solve this!\u201d It then shows that x \u2013 2(7 \u2013 2x) = 6 becomes x \u2013 14 + 4x = 6 which becomes 5x = 20. Thus x = 4.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001c_img_new.jpg\" alt=\"The third row says, \u201cStep 3: Solve the resulting equation.\u201d Then \u201cNow we have an equation with just 1 variable. WE know how to solve this!\u201d It then shows that x \u2013 2(7 \u2013 2x) = 6 becomes x \u2013 14 + 4x = 6 which becomes 5x = 20. Thus x = 4.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345328036\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4. Substitute the solution in Step 3 into one of the original quaitons to find the other variable.\u201d Then, \u201cWe\u2019ll use the first equation and replace x with 4.\u201d Then it shows that 2x + y = 7 becomes 2(4) + y = 7. This becomes 8 + y = 7, and thus y = \u22121.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4. Substitute the solution in Step 3 into one of the original quaitons to find the other variable.\u201d Then, \u201cWe\u2019ll use the first equation and replace x with 4.\u201d Then it shows that 2x + y = 7 becomes 2(4) + y = 7. This becomes 8 + y = 7, and thus y = \u22121.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345254426\" data-type=\"media\" data-alt=\"The fifth row reads, \u201cStep 5. Write the solution as an ordered pair.\u201d Then \u201cThe ordered air is (x, y).\u201d Then (4, \u22121).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001e_img_new.jpg\" alt=\"The fifth row reads, \u201cStep 5. Write the solution as an ordered pair.\u201d Then \u201cThe ordered air is (x, y).\u201d Then (4, \u22121).\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345287958\" data-type=\"media\" data-alt=\"The sixth row reads, \u201cStep 6. Check that the order pair is a solution to both original equations.\u201d Then, \u201cSubstitute (4, \u22121) into both equations and make sure they are both true.\u201d It then shows that 2x + y = 7 becomxe 2(4) + \u22121 = 7, and thus 7 = 7. It also shows that x \u2013 2y = 6 becomes 4 \u2013 2(\u22121) = 6, and thus 6\u22126. It also states, \u201cBoth equations are ture. (4, \u22121) is the solution to the system.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001f_img_new.jpg\" alt=\"The sixth row reads, \u201cStep 6. Check that the order pair is a solution to both original equations.\u201d Then, \u201cSubstitute (4, \u22121) into both equations and make sure they are both true.\u201d It then shows that 2x + y = 7 becomxe 2(4) + \u22121 = 7, and thus 7 = 7. It also shows that x \u2013 2y = 6 becomes 4 \u2013 2(\u22121) = 6, and thus 6\u22126. It also states, \u201cBoth equations are ture. (4, \u22121) is the solution to the system.\u201d\" data-media-type=\"image\/jpeg\"><\/span>\n\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 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341852430\" data-type=\"problem\">\n<p id=\"fs-id1168345417415\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}-2x+y=-11\\hfill \\\\ x+3y=9\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168341961510\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345408042\">\\(\\left(6,1\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">How to solve a system of equations by substitution.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169753231587\" class=\"stepwise\" type=\"1\">\n \t<li>Solve one of the equations for either variable.<\/li>\n \t<li>Substitute the expression from Step 1 into the other equation.<\/li>\n \t<li>Solve the resulting equation.<\/li>\n \t<li>Substitute the solution in Step 3 into one of the original equations to find the other variable.<\/li>\n \t<li>Write the solution as an ordered pair.<\/li>\n \t<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345216137\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345340004\">If one of the equations in the system is given in slope\u2013intercept form, Step 1 is already done!<\/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 id=\"fs-id1168341960594\" data-type=\"problem\">\n<p id=\"fs-id1168345203647\">Solve the system by substitution.<\/p>\n<p id=\"fs-id1169751032452\">\\(\\left\\{\\begin{array}{c}x+y=-1\\hfill \\\\ y=x+5\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345284532\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345254082\">The second equation is already solved for <em data-effect=\"italics\">y<\/em>. We will substitute the expression in place of <em data-effect=\"italics\">y<\/em> in the first equation.<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167836448172\" style=\"width: 100%;\" summary=\"In this figure we have two columns that work out the solution. In the top of the right column is the system of equation \u201cx plus y equals negative 1\u201d and \u201cy equals x plus 5.\u201d The next line down on the left reads, \u201cThe second equation is already solved for y. We will substitute into the first equation. Next to this is \u201cy equals x plus 5. The \u201cx plus 5\u201d is within a red circle with an arrow pointing to the \u201cy\u201d in the next line, which indicates that \u201cx plus 5\u201d will be the substitute for \u201cy\u201d. The next line on the left reads, \u201cReplace the \u201cy\u201d with \u201cx plus 5\u201d. Then, \u201cSolve the resulting equation for \u201cx\u201d. Next to this on the right is the equation \u201cx plus x plus 5 equals negative 1.\u201d Under this is is \u201c2x plus 5 equals negative 1\u201d, under this is\u201d2x equals negative 6\u201d and under this is \u201cx equals negative 3\u201d. The \u201cequals negative 3\u201d has a red circle around it with an arrow pointing to the \u201cx\u201d in the expression below. The next line down on the left reads, \u201cSubstitute x plus negative 3 into \u201cy equals x plus 5 to find y\u201d. To the right of this is the expression \u201cy equals x plus 5\u201d. Under this is \u201cy equals negative 3 plus 5\u201d, and under this is \u201cy equals 2\u201d. The next line down on the left reads, \u201cthe ordered pair is negative 3, 2\u201d in parentheses. Below this it reads, \u201cCheck the ordred pair in both equations.\u201d Below this we have both sets of equations. On the left if x plus y equals negative 1, under this negative 3 plus 2 equals negative one. A question mark is above the equal sign. Under this is negative 1 equals negative 1 with a checkmark beside it. The equation to the right of this reads, y equals x plus 5. Undr this is 2 equals negative 3 plus 5 with a questions mark above the equal sign. Under this is 2 equals 2, with a checkmark beside it. In the right column, the last line reads, \u201cthe solution is ordered pair negative 3 and 2.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836700889\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">The second equation is already solved for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\">\n<\/span>We will substitute into the first equation.<\/td>\n<td style=\"width: 295px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"bottom\">Replace the <em data-effect=\"italics\">y<\/em> with <em data-effect=\"italics\">x<\/em> + 5.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836700881\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836329838\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836575027\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836605298\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"bottom\">Substitute <em data-effect=\"italics\">x<\/em> = \u22123 into <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">x<\/em> + 5 to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836492201\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836534698\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">The ordered pair is (\u22123, 2).<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836618806\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill x+y&amp; =\\hfill &amp; -1\\hfill \\\\ \\hfill -3+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 y&amp; =\\hfill &amp; x+5\\hfill \\\\ \\hfill 2&amp; \\stackrel{?}{=}\\hfill &amp; -3+5\\hfill \\\\ \\hfill 2&amp; =\\hfill &amp; 2\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 295px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-align=\"center\" data-valign=\"top\">The solution is (\u22123, 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 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345448675\" data-type=\"problem\">\n<p id=\"fs-id1168345374070\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}x+y=6\\hfill \\\\ y=3x-2\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345541402\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345254372\">\\(\\left(2,4\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345451707\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345557251\" data-type=\"exercise\">\n<div id=\"fs-id1168345448675\" data-type=\"problem\">\n\n<span style=\"text-align: initial; font-size: 14pt;\">If the equations are given in standard form, we\u2019ll need to start by solving for one of the variables. In this next example, we\u2019ll solve the first equation for <\/span><em style=\"text-align: initial; font-size: 14pt;\" data-effect=\"italics\">y.<\/em>\n\n<\/div>\n<\/div>\n<\/div>\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-id1168345746371\" data-type=\"problem\">\n<p id=\"fs-id1168345452943\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}3x+y=5\\hfill \\\\ 2x+4y=-10\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345415462\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345285867\">We need to solve one equation for one variable. Then we will substitute that expression into the other equation.<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167829753382\" style=\"width: 100%;\" summary=\"This figure show two equations: 3x + y = 5 and 2x + 4y = -10. It indicates, \u201cSolve for y.\u201d and shows 3x + y = 5 becomes y = -3x + 5. Then it says, \u201cSubstitute into the other equation. Relace the y with -3x + 5. Solve the resulting equation for x.\u201d It then shows that 2x + 4y = -10 becomes 2x + 4(-3x +5) = -10. This then becomes 2x -12x +20 = -10 which becomes -10x + 20 = -10. This becomes -10x = -30 which means x = 3. The figure then says, \u201cSubstitute x = 3 into 3x + y = 5 to find y.\u201d Thus 3x + y = 5 becomes 3(3) + y = 5. This becomes 9 + y = 5 which means y = -4. The figure then indicates, \u201cThe ordered pair is (3, -4). The figure then instructs, \u201cCheck the ordered pari in both equations:\u201d Thus 3x + y = 5 becomes 3 times 3 + (-4) = 5. This is 9 \u2013 4 = 5 which is 5 = 5. 2x + 4y = -10 becomes 2 times 3 + 4(-4) = -10 which means 6 \u2013 16 = -10 and thus -10 = -10. The figure then says, \u201cThe solution is (3, -4).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"bottom\">Solve for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute into the other equation.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836318910\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Replace the <em data-effect=\"italics\">y<\/em> with \u22123<em data-effect=\"italics\">x<\/em> + 5.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167832994444\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836544330\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829719558\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829893884\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"bottom\">Substitute <em data-effect=\"italics\">x<\/em> = 3 into 3<em data-effect=\"italics\">x<\/em> + <em data-effect=\"italics\">y<\/em> = 5 to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836556269\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\"><\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829908075\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167833136578\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">The ordered pair is (3, \u22124).<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829696105\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill 3x+y&amp; =\\hfill &amp; 5\\hfill \\\\ \\hfill 3 \\cdot 3+\\left(-4\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 5\\hfill \\\\ \\hfill 9-4&amp; \\stackrel{?}{=}\\hfill &amp; 5\\hfill \\\\ \\hfill 5&amp; =\\hfill &amp; 5\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill 2x+4y&amp; =\\hfill &amp; -10\\hfill \\\\ \\hfill 2 \\cdot 3+4\\left(-4\\right)&amp; =\\hfill &amp; -10\\hfill \\\\ \\hfill 6-16&amp; \\stackrel{?}{=}\\hfill &amp; -10\\hfill \\\\ \\hfill -10&amp; =\\hfill &amp; -10\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 46.5347%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\"><\/td>\n<td style=\"width: 46.5347%;\" data-align=\"center\" data-valign=\"top\">The solution is (3, \u22124).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345746702\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345426936\" data-type=\"exercise\">\n<div id=\"fs-id1168345384384\" data-type=\"problem\">\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-id1168345384384\" data-type=\"problem\">\n<p id=\"fs-id1168341864330\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}4x+y=2\\hfill \\\\ 3x+2y=-1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345250252\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345213782\">\\(\\left(1,-2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345386232\">In example 3, it was easiest to solve for <em data-effect=\"italics\">y<\/em> in the first equation because it had a coefficient of 1. In the next example it will be easier to solve for <em data-effect=\"italics\">x<\/em>.<\/p>\n\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-id1168345436528\" data-type=\"problem\">\n<p id=\"fs-id1168341840994\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}x-2y=-2\\hfill \\\\ 3x+2y=34\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345256708\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345253030\">We will solve the first equation for \\(x\\) and then substitute the expression into the second equation.<\/p>\n\n<table id=\"fs-id1167829596505\" style=\"width: 100%;\" summary=\"This figure says, \u201cSolve for x.\u201d Thus x \u2013 27 = -2 becomes x = 2y -2. The figure then instructs, \u201cSubstitute into the other equations Replace x with 2y -2. Solve for the resulting equation for y.\u201d Thus 3x + 2y = 34 becomes 3(2y -2) +2y = 34. This then becomes 6y \u2013 6 +2y = 34. This becomes 8y \u2013 6 = 34 or 8y = 40. Thus y = 5. The figure then instructs, \u201cSubstitute y = 5 inot x \u2013 2y = -2 to find x.\u201d Thus x \u2013 2y = -2 becomes x \u2013 2 times 5 = -2. This becomes x \u2013 10 = -2 which menas x = 8. The figure then instructs, \u201cThe ordered pair is (8, 5). Check the ordered pair in both equations:\u201d It then shows that x \u2013 2y = -2 becomes 8 \u2013 2 times 5 = -2. This becomes 8 \u2013 10 = -2 and thus -2 = -2. This also shows that 3x + 2y = 34 becomes 3 times 8 plus 2 times 5 = 34. This becomes 24 + 10 = 34 and thus 34 = 34. The figure then states, \u201cThe solution is (8, 5).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 297.062px;\"><\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167835384764\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Solve for <em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute into the other equation.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167836361817\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Replace the <em data-effect=\"italics\">x<\/em> with 2<em data-effect=\"italics\">y<\/em> \u2212 2.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167833031496\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167836287768\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"middle\"><span data-type=\"newline\">\n<\/span>Substitute <em data-effect=\"italics\">y<\/em> = 5 into <em data-effect=\"italics\">x<\/em> \u2212 2<em data-effect=\"italics\">y<\/em> = \u22122 to find <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167833136523\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167832945647\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836503161\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829691575\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836521554\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836388843\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">The ordered pair is (8, 5).<\/td>\n<td style=\"width: 178.156px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill x-2y&amp; =\\hfill &amp; -2\\hfill \\\\ \\hfill 8-2 \\cdot 5&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill \\\\ \\hfill 8-10&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill \\\\ \\hfill -2&amp; =\\hfill &amp; -2\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill 3x+2y&amp; =\\hfill &amp; 34\\hfill \\\\ \\hfill 3 \\cdot 8+2 \\cdot 5&amp; \\stackrel{?}{=}\\hfill &amp; 34\\hfill \\\\ \\hfill 24+10&amp; \\stackrel{?}{=}\\hfill &amp; 34\\hfill \\\\ \\hfill 34&amp; =\\hfill &amp; 34\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 178.156px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\"><\/td>\n<td style=\"width: 178.156px;\" data-align=\"center\" data-valign=\"top\">The solution is (8, 5).<\/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-id1168345302419\" data-type=\"problem\">\n<p id=\"fs-id1168345452325\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}x-5y=13\\hfill \\\\ 4x-3y=1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345231497\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345346380\">\\(\\left(-2,-3\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345301408\">When both equations are already solved for the same variable, it is easy to substitute!<\/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-id1168345541944\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345230089\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}y=-2x+5\\hfill \\\\ y=\\frac{1}{2}x\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345742979\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345453561\">Since both equations are solved for\u00a0<em data-effect=\"italics\">y<\/em>, we can substitute one into the other.<\/p>\n\n<table id=\"fs-id1167829714127\" class=\"unnumbered unstyled can-break\" summary=\"This figure instructs, \u201cSubstitute (1\/2)x for y in the first equation. Replace the y with (1\/2)x. Solve the resulting equation. Start by clearing the fraction. Solve for x.\u201d Thus y = (1\/2)x substituts into y = -2x +5. This becomes (1\/2)x = -2x +5. Then 2((1\/2)x) = 2(-2x + 5) which becomes x = -4x +10. This becomes 5x =10 which means x = 2. The figure then instructs, \u201cSubstitute x = 2 into y = (1\/2)x to find y. Thus y = (1\/2)x becomes y = (1\/2) times 2, and thus y = 1. The figure then says, \u201cThe ordered pair is (2, 1). It then instructs, \u201cCheck the ordered pair in both equations:\u201d Thus y = (1\/2)x becomes 1 = 1\/2 times 2, and thus 1 = 1. Y = -2x +5 becomes 1 = -2 times 2 + 5. This becomes 1 = -4 +5 and thus 1 = 1. The figure states, \u201cThe solution is (2, 1).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"bottom\">Substitute \\(\\frac{1}{2}x\\) for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167829746913\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with \\(\\frac{1}{2}x.\\)<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836329209\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Solve the resulting equation. Start<span data-type=\"newline\">\n<\/span>by clearing the fraction.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836548764\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167833086359\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\"><\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836619846\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\">Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 2 into\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \\(\\frac{1}{2}x\\) to find\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167824735799\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167824732580\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836352956\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">The ordered pair is (2,1).<\/td>\n<td style=\"width: 262.531px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; \\frac{1}{2}x\\hfill \\\\ \\hfill 1&amp; \\stackrel{?}{=}\\hfill &amp; \\frac{1}{2}\\cdot 2\\hfill \\\\ \\hfill 1&amp; =\\hfill &amp; 1\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; -2x+5\\hfill \\\\ \\hfill 1&amp; \\stackrel{?}{=}\\hfill &amp; -2 \\cdot 2+5\\hfill \\\\ \\hfill 1&amp; =\\hfill &amp; -4+5\\hfill \\\\ \\hfill 1&amp; =\\hfill &amp; 1\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 262.531px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\"><\/td>\n<td style=\"width: 262.531px;\" data-align=\"center\" data-valign=\"top\">The solution is (2,1).<\/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-id1168345667581\" data-type=\"problem\">\n<p id=\"fs-id1168345216253\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}y=3x-16\\hfill \\\\ y=\\frac{1}{3}x\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345256614\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345199012\">\\(\\left(6,2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341961701\">Be very careful with the signs in the next example.<\/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-id1168345293219\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345727316\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}4x+2y=4\\hfill \\\\ 6x-y=8\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345424365\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345300933\">We need to solve one equation for one variable. We will solve the first equation for\u00a0<em data-effect=\"italics\">y<\/em>.<\/p>\n\n<table id=\"fs-id1167836625241\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSolve the first equation for y.\u201d Thus, 4x + 2y = 4 becomes 2y = -4x + 4. This then becomes y = -2x + 2. The figure then instructs, \u201cSubstitute -2x + 2 for y in the second equation. Replace the y with -2x +2. Solve the equation for x.\u201d Thus 6x \u2013 y = 8 becomes 6x \u2013 (-2x + 2) = 8. This becomes 6x + 2x -2) = 8. This becomes 8x \u2013 2 = 8. Thus 8x = 10 and x = 5\/4. The figure then instructs, \u201cSubstitute x = 5\/4 into 4x +2y = 4 to find y. Thus 4x +2y = 4 becomes 4(5\/4) + 2y = 4. This becomes 5 + 2y = 4 or 2y = -1, and thus y = -1\/2. The figure then says, \u201cThe ordered pair is (5\/4, -1\/2).\u201d It then instructs, \u201cCheck the ordered pair in both equations:\u201d Thus 4x +2y = 4 becomes 4(5\/4) + 2(-1\/2) = 4. This becomes 5 \u2013 1 = 4 and thus 4 = 4. 6x \u2013 y = 8 becomes 6(5\/4) \u2013 (-1\/2) = 8. This becomes 15\/4 \u2013(-1\/2) = 8. Thus 16\/2 = 8 and 8 = 8. The figure then says, \u201cThe solution is (5\/4, -1\/2).\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836628457\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Solve the first equation for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836368137\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"bottom\">Substitute \u22122<em data-effect=\"italics\">x<\/em>\u00a0+ 2 for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the second equation.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836698748\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with \u22122<em data-effect=\"italics\">x<\/em>\u00a0+ 2.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"bottom\"><span id=\"fs-id1167836388313\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Solve the equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"bottom\"><span id=\"fs-id1167836359959\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836299914\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836568022\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\"><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute \\(x=\\frac{5}{4}\\) into 4<em data-effect=\"italics\">x<\/em>\u00a0+ 2<em data-effect=\"italics\">y<\/em>\u00a0= 4 to find\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836549263\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829693698\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836310171\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829596793\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836315024\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">The ordered pair is \\(\\left(\\frac{5}{4},-\\frac{1}{2}\\right).\\)<\/td>\n<td style=\"width: 45.9406%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Check the ordered pair in both equations.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill 4x+2y&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill 4\\left(\\frac{5}{4}\\right)+2\\left(-\\frac{1}{2}\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ \\hfill 5-1&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ \\hfill 4&amp; =\\hfill &amp; 4\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{ccc}\\hfill 6x-y&amp; =\\hfill &amp; 8\\hfill \\\\ \\hfill 6\\left(\\frac{5}{4}\\right)-\\left(-\\frac{1}{2}\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 8\\hfill \\\\ \\hfill \\frac{15}{4}-\\left(-\\frac{1}{2}\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 8\\hfill \\\\ \\hfill \\frac{16}{2}&amp; \\stackrel{?}{=}\\hfill &amp; 8\\hfill \\\\ \\hfill 8&amp; =\\hfill &amp; 8\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\hfill \\end{array}\\)<\/td>\n<td style=\"width: 45.9406%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-align=\"center\" data-valign=\"top\">The solution is \\(\\left(\\frac{5}{4},-\\frac{1}{2}\\right).\\)<\/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-id1168345560310\" data-type=\"problem\">\n<p id=\"fs-id1168345436701\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}x-4y=-4\\hfill \\\\ -3x+4y=0\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345416943\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345370970\">\\(\\left(2,\\frac{3}{2}\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345744719\">In the next example, it will take a little more work to solve one equation for <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>.<\/p>\n\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 style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345290615\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}4x-3y=6\\hfill \\\\ 15y-20x=-30\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345285072\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345424353\">We need to solve one equation for one variable. We will solve the first equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/p>\n\n<table id=\"fs-id1167829692301\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSolve the first equation for x.\u201d Thus 4x -3y = 6 becomes 4x = 3y + 6. This becomes x = (3\/4)y + 3\/2. The figure then instructs, \u201cSubstitute (3\/4)y + 3\/2 for x in the section equation. Replace the x with (3\/4)y + 3\/2. Solve for y.\u201d Thus 15y -20x = -30 becomes 15y \u2013 20((3\/4)y + 3\/2) = -30. Then 15y -15y -30 = -30 becomes 0 -30 = -30 or 0 = 0.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167825998321\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"top\">Solve the first equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836665073\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"bottom\">Substitute \\(\\frac{3}{4}y+\\frac{3}{2}\\) for\u00a0<em data-effect=\"italics\">x<\/em>\u00a0in the second equation.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836352297\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">x<\/em>\u00a0with \\(\\frac{3}{4}y+\\frac{3}{2}.\\)<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836549360\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"bottom\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836731170\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167829859295\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836330142\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345242251\">Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.<\/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 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345408367\" data-type=\"problem\">\n<p id=\"fs-id1168345373805\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}2x-3y=12\\hfill \\\\ -12y+8x=48\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345439980\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345196602\">infinitely many solutions<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341923833\">Look back at the equations in example 6. Is there any way to recognize that they are the same line?<\/p>\n<p id=\"fs-id1168345544175\">Let\u2019s see what happens in the next example.<\/p>\n\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-id1168345473672\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168341955763\">Solve the system by substitution. \\(\\left\\{\\begin{array}{c}5x-2y=-10\\hfill \\\\ y=\\frac{5}{2}x\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n&nbsp;\n<div id=\"fs-id1168345695456\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345273788\">The second equation is already solved for\u00a0<em data-effect=\"italics\">y<\/em>, so we can substitute for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/p>\n\n<table id=\"fs-id1167836635488\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSubstitue f for y in the first equation. Replace the y with (5\/2)x. Solve for x.\u201d Thus y = (5\/2)x is substituted is not 5x minus 2((5\/2)x) = negative 10. This becomes 5x \u2013 5x = negative 10 or 0 = negative 10.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"bottom\"><span id=\"fs-id1167836704941\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with \\(\\frac{5}{2}x.\\)<\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\"><span id=\"fs-id1167836331212\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"bottom\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\"><span id=\"fs-id1167836361374\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\"><\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\" data-align=\"right\"><span id=\"fs-id1167836323046\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168341863633\">Since 0 = \u221210 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.<\/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 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nSolve the system by substitution. \\(\\left\\{\\begin{array}{c}3x+2y=9\\hfill \\\\ y=-\\frac{3}{2}x+1\\hfill \\end{array}\\)\n<div id=\"fs-id1168345210433\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345210581\">\u00a0 \u00a0no solution<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341864163\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Substitution<\/h1>\n<p id=\"fs-id1168345373909\">We\u2019ll copy here the problem solving strategy we used in the last sub chapter for solving systems of equations. Now that we know how to solve systems by substitution, that\u2019s what we\u2019ll do in Step 5.<\/p>\n\n<div id=\"fs-id1168345574727\" 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<ol id=\"fs-id1169746619671\" 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<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341923605\">Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example\u2014how would you have done it with just one variable?<\/p>\n\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-id1168345414503\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345414731\">The sum of two numbers is zero. One number is nine less than the other. Find the numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345744922\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836529197\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two numbers. Step 3. Name what we are looking for. Let n = the first number and let m = the second number. Step 4. Translate into a system of equations. The sum of two numbers is zero. N + m = 0. One number is nine less than the other or n = m -9. The system is: n + m = 0 and n = m \u2013 9. Step 5. Solve the system of equations. We will use substitution since the second equation is solved for ns. Substitute m -9 for n in the first equations. Solve for m.\u201d Thus n = m \u2013 9 is substituted into n + m = 0 and becomes m \u2013 9 + m = 0 or 2m \u2013 9 = 0. Thus 2m = 9 and m = 9\/2. The figure then instructs, \u201cSubstitute m = 9\/2 into the second equation and then solve for n.\u201d Thus n = m \u2013 9 bexomes m = 9\/2 \u2013 9 or m = 9\/2 \u2013 18\/2. Thus n = -9\/2. The figure then instructs, \u201cStep 6. Check the answer in the problem. Does these numbers make sense in the problem? We will leave this to you!\u201d Step 7. Answer the question. The numbers are 9\/2 and \u2013 9\/2.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">We are looking for two numbers.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">\u00a0Let \\(n=\\) the first number<span data-type=\"newline\">\n<\/span>Let \\(m=\\) the second number<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The sum of two numbers is zero.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829844007\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">One number is nine less than the other.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836613186\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836551442\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of<span data-type=\"newline\">\n<\/span>equations. We will use substitution<span data-type=\"newline\">\n<\/span>since the second equation is solved<span data-type=\"newline\">\n<\/span>for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">m<\/em>\u00a0\u2212 9 for\u00a0<em data-effect=\"italics\">n<\/em>\u00a0in the first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836415079\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">m<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836294980\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009f_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-id1167833139030\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009g_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-id1167836327185\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute \\(m=\\frac{9}{2}\\) into the second equation<span data-type=\"newline\">\n<\/span>and then solve for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"bottom\"><span id=\"fs-id1167833058872\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009i_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-id1167836555974\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009j_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-id1167836688249\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009k_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-id1167833139392\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">Do these numbers make sense in<span data-type=\"newline\">\n<\/span>the problem? We will leave this to you!<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The numbers are \\(\\frac{9}{2}\\) and \\(-\\frac{9}{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 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345275137\" data-type=\"problem\">\n<p id=\"fs-id1168345241744\">The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345742616\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345461982\">The numbers are 3 and 7.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341916379\">In the next example, we\u2019ll use the formula for the perimeter of a rectangle, <em data-effect=\"italics\">P<\/em> = 2<em data-effect=\"italics\">L<\/em> + 2<em data-effect=\"italics\">W<\/em>.<\/p>\n\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-id1168345275902\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345560167\">The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345252894\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836686435\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem.\u201d There is a drawing of a rectangle with the width (w) and length (l) marked. The figure instructs, \u201cStep 2. Identify what you are looking for. We are looking for the length and width. Step 3. Name what we are looking for. Let L = the length, W = the width. Step 4. Translate into a system of equations. The perimeter of the rectangle is 88. 2L + 2W = 88. The length is five more than twice the width. L = 2W + 5. The system is 2L + 2W = 88 and L = 2W + 5. Step 5. Solve the system of equations. We will use substation since the second equation is solved of L. Substitute 2W + 5 for L in the first equations. Solve for W.\u201d Thus 2L + 2W = 88 becomes 2(2W + 5) + 2W = 88 which becomes 4W + 10 + 2W = 22. This becomes 6W +10 = 88 or 6W = 78 or W = 13. The figure then instructs, \u201cSubstitute W = 13 into the second equation and then solve for L. L = 2W + 5 becomes L = 2 times 13 + 5. Thus L = 31. The figure then instructs, \u201cStep 6. Check that answer in the problem. Does a rectangle with length 31 and width 18 have perimeter 88? Yes. Step 7. Answer the question. The length is 31 and the width is 13.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829921210\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the length and width.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let \\(L=\\) the length<span data-type=\"newline\">\n<\/span>\u2003\u00a0\\(W=\\) the width<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The perimeter of a rectangle is 88.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">\u2003\u2003\u00a0\u00a02<em data-effect=\"italics\">L<\/em>\u00a0+ 2<em data-effect=\"italics\">W<\/em>\u00a0=\u00a0<em data-effect=\"italics\">P<\/em><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836415488\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">The length is five more than twice the width.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836294986\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829743944\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\">\n<\/span>We will use substitution since the second<span data-type=\"newline\">\n<\/span>equation is solved for\u00a0<em data-effect=\"italics\">L<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute 2<em data-effect=\"italics\">W<\/em>\u00a0+ 5 for\u00a0<em data-effect=\"italics\">L<\/em>\u00a0in the first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836362612\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">W<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836363048\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010h_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-id1167829739841\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010i_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-id1167836546317\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010j_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-id1167832950892\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">W<\/em>\u00a0= 13 into the second<span data-type=\"newline\">\n<\/span>equation and then solve for\u00a0<em data-effect=\"italics\">L<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836558356\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010l_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-id1167833379409\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010m_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-id1167836507472\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010n_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">Does a rectangle with length 31 and width<span data-type=\"newline\">\n<\/span>13 have perimeter 88? Yes.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the equation.<\/td>\n<td data-valign=\"top\">The length is 31 and the width is 13.<\/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 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341952485\" data-type=\"problem\">\n<p id=\"fs-id1168345274903\">The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345724938\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345297385\">The length is 12 and the width is 8.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345368011\">For the next example we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.<\/p>\n\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-id1168345745360\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345579068\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345254941\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1168345483406\">We will draw and label a figure.<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167836616035\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1: Read the problem.\u201d This is a right triangle drawn with one angle labeled a and one angle labeled b. The figure further instructs, \u201cStep 2: Identify what we are looking for. We are looking for measures of the angles. Step 3: Name what we are looking for. Let a = the measure of the first angel and b = the measure of the second angle. Step 4: Translate into a system of equations. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other angle.\u201d Thus a = 3b + 10. \u201cThe sume of the measure of tha angles of a triangle is 180.\u201d Thus, a + b + 90 = 180. \u201cThe system is: a = 3b + 10, a + b + 90 = 180. Step 5. Solve the system of equations. We will use substitution since the first equation is solved for a. Substitute 3b +10 for a in the second equation. Solve for b.\u201d thus a + b + 90 = 180 becomes (3b + 10) + b + 90 = 180. This becomes 4b +100 = 18 or 4b = 80. Thus b = 20. The figure then instructs, \u201cSubstitute b = 20 into the first equation and then solve for a.\u201d Thus a = 3b + 10 becomes a = 3 times 20 +10 or a = 70. The figure then instructs, \u201cStep 6: Check the answer in the problem. We will leave this to you! Step 7: Answer the question. The measures of the small angels are 20 and 70.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836300370\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the measures of the angles.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let \\(a=\\) the measure of the 1<sup>st<\/sup>\u00a0angle<span data-type=\"newline\">\n<\/span>\\(\\phantom{\\rule{1.5em}{0ex}}b=\\) the measure of the 2<sup>nd<\/sup>\u00a0angle<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The measure of one of the small angles<span data-type=\"newline\">\n<\/span>of a right triangle is ten more than three<span data-type=\"newline\">\n<\/span>times the measure of the other small angle.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829720983\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">The sum of the measures of the angles of<span data-type=\"newline\">\n<\/span>a triangle is 180.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836416553\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829784979\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\">\n<\/span>We will use substitution since the first<span data-type=\"newline\">\n<\/span>equation is solved for\u00a0<em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836550173\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute 3<em data-effect=\"italics\">b<\/em>\u00a0+ 10 for\u00a0<em data-effect=\"italics\">a<\/em>\u00a0in the<span data-type=\"newline\">\n<\/span>second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836684101\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Solve for\u00a0<em data-effect=\"italics\">b<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836523331\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011h_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-id1167829594592\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011i_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-id1167836560598\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute\u00a0<em data-effect=\"italics\">b<\/em>\u00a0= 20 into the first<span data-type=\"newline\">\n<\/span>equation and then solve for\u00a0<em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836525449\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167829711937\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">We will leave this to you!<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The measures of the small angles are<span data-type=\"newline\">\n<\/span>20 and 70.<\/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 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345287011\" data-type=\"note\">\n<div id=\"fs-id1168345541526\" data-type=\"exercise\">\n<div id=\"fs-id1168345432468\" data-type=\"problem\">The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345250240\" data-type=\"note\">\n<div id=\"fs-id1168345551489\" data-type=\"exercise\">\n<div id=\"fs-id1168345441083\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345324882\">The measure of the angles are 22 degrees and 68 degrees.<\/p>\n\n<\/details><\/div>\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-id1168345388247\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n\nHeather has been offered two options for her salary as a trainer at the gym. Option A would pay her \\$25,000 plus \\$15 for each training session. Option B would pay her \\$10,000 + \\$40 for each training session. How many training sessions would make the salary options equal?\n\n<\/div>\n<div id=\"fs-id1168345432359\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836630339\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what you are looking for. We are looking for the number of training sessions that would make the pay equal. Step 3. Name what we are looking for. Let s = Heather\u2019s salary, n = the number of training session. Step 4. Translate into a system of equations. Option A would pay her ?25,000 plus ?15 for each taining session. S = 25000 + 15n; Option would pay her ?10,000 + ?40 for each training session. S = 10,000 + 40n. The system is s = 25,000 +15n, s = 10,000 + 40n. Step 5. Solve the system of equations. We will use sbustitiution. Substitute 25,000 + 15n for s in the second equation. Solve or n.\u201d Thus s = 10,000 +40n becomes 25,000 + 15 n = 10,000 + 40n. This becomes 25,000 = 10,000 + 25n which becomes 15,000 = 25n or n = 600. The figure then instructs, \u201cStep 6. Check the answer. Are 600 training session a year reasonable? Are the two options equal when n = 600? Step 7. Answer the questions. The salary options would be equal for 600 training sessions.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the number of training sessions<span data-type=\"newline\">\n<\/span>that would make the pay equal.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let \\(s=\\) Heather\u2019s salary.<span data-type=\"newline\">\n<\/span>\\(\\phantom{\\rule{1.5em}{0ex}}n=\\) the number of training sessions<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">Option A would pay her \\$25,000 plus \\$15<span data-type=\"newline\">\n<\/span>for each training session.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167832999774\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Option B would pay her \\$10,000 + \\$40<span data-type=\"newline\">\n<\/span>for each training session<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836660076\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836622610\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\">\n<\/span>We will use substitution.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836296961\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute 25,000 + 15<em data-effect=\"italics\">n<\/em>\u00a0for\u00a0<em data-effect=\"italics\">s<\/em>\u00a0in the second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836533652\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833060219\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012g_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-id1167836623222\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012h_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-id1167836635265\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td data-valign=\"top\">Are 600 training sessions a year reasonable?<span data-type=\"newline\">\n<\/span>Are the two options equal when\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= 600?<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The salary options would be equal for 600 training sessions.<\/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 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345415652\" data-type=\"problem\">\n<p id=\"fs-id1168345240744\">Geraldine has been offered positions by two insurance companies. The first company pays a salary of \\$12,000 plus a commission of \\$100 for each policy sold. The second pays a salary of \\$20,000 plus a commission of \\$50 for each policy sold. How many policies would need to be sold to make the total pay the same?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345386624\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345345938\">There would need to be 160 policies sold to make the total pay the same.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"text-align: initial; font-size: 14pt;\">Access these online resources for additional instruction and practice with solving systems of equations by substitution.<\/span>\n<div id=\"fs-id1168345432571\" class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1168345294286\" data-display=\"block\">\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25SolvingLinear\">Instructional Video-Solve Linear Systems by Substitution<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25Substitution\">Instructional Video-Solve by Substitution<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168341863466\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">Solve a system of equations by substitution<\/strong>\n<ol id=\"fs-id1169753334202\" class=\"stepwise\" type=\"1\">\n \t<li>Solve one of the equations for either variable.<\/li>\n \t<li>Substitute the expression from Step 1 into the other equation.<\/li>\n \t<li>Solve the resulting equation.<\/li>\n \t<li>Substitute the solution in Step 3 into one of the original equations to find the other variable.<\/li>\n \t<li>Write the solution as an ordered pair.<\/li>\n \t<li>Check that the ordered pair is a solution to both original equations.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1 data-type=\"title\">4.2 Exercise Set<\/h1>\n<p id=\"fs-id1169750685787\">In the following exercises, solve the systems of equations by substitution.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=-4\\hfill \\\\ 3x-2y=-6\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x-2y=-5\\hfill \\\\ 2x-3y=-4\\hfill \\end{array}\\)<\/li>\n \t<li><span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\">\\(\\left\\{\\begin{array}{c}5x-2y=-6\\hfill \\\\ y=3x+3\\hfill \\end{array}\\)<\/span><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+3y=3\\hfill \\\\ y=\\text{\u2212}x+3\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+5y=1\\hfill \\\\ y=\\frac{1}{3}x-2\\hfill \\end{array}\\)<\/li>\n \t<li><span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\">\\(\\left\\{\\begin{array}{c}3x-2y=6\\hfill \\\\ y=\\frac{2}{3}x+2\\hfill \\end{array}\\)<\/span><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=10\\hfill \\\\ -x+y=-5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x+y=1\\hfill \\\\ -4x+y=15\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+3y=1\\hfill \\\\ 3x+5y=-5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=5\\hfill \\\\ x-2y=-15\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=-2x-1\\hfill \\\\ y=-\\frac{1}{3}x+4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=2x-8\\hfill \\\\ y=\\frac{3}{5}x+6\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}4x+2y=8\\hfill \\\\ 8x-y=1\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}15x+2y=6\\hfill \\\\ -5x+2y=-4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=3x\\hfill \\\\ 6x-2y=0\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+16y=8\\hfill \\\\ -x-8y=-4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=-4x\\hfill \\\\ 4x+y=1\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y=\\frac{7}{8}x+4\\hfill \\\\ -7x+8y=6\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<p id=\"fs-id1169746349628\">In the following exercises, translate to a system of equations and solve.<\/p>\n\n<ol start=\"19\">\n \t<li>The sum of two numbers is 15. One number is 3 less than the other. Find the numbers<\/li>\n \t<li>The sum of two numbers is \u221226. One number is 12 less than the other. Find the numbers.<\/li>\n \t<li>The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.<\/li>\n \t<li>The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.<\/li>\n \t<li>The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/li>\n \t<li>The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.<\/li>\n \t<li>Jackie has been offered positions by two cable companies. The first company pays a salary of \\$ 14,000 plus a commission of \\$100 for each cable package sold. The second pays a salary of \\$20,000 plus a commission of \\$25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?<\/li>\n \t<li>Mitchell currently sells stoves for company A at a salary of \\$12,000 plus a \\$150 commission for each stove he sells. Company B offers him a position with a salary of \\$24,000 plus a \\$50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?<\/li>\n \t<li>\n<p id=\"fs-id1168345550140\">Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system \\(\\left\\{\\begin{array}{c}56s=70t\\hfill \\\\ s=t+\\frac{1}{2}\\hfill \\end{array}\\).<\/p>\n\n<ol type=\"a\">\n \t<li>for \\(t\\) to find out how long it will take Tina to catch up to Stephanie.<\/li>\n \t<li>what is the value of \\(s\\), the number of hours Stephanie will have driven before Tina catches up to her?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1168345729576\" data-type=\"exercise\">\n<div id=\"fs-id1168345729579\" data-type=\"problem\">\n<p id=\"fs-id1168345729581\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; background-color: initial;\">Answers:<\/span><\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\\(\\left(-2,0\\right)\\)<\/li>\n \t<li>\\(\\left(7,6\\right)\\)<\/li>\n \t<li>\\(\\left(0,3\\right)\\)<\/li>\n \t<li>\\(\\left(6,-3\\right)\\)<\/li>\n \t<li>\\(\\left(3,-1\\right)\\)<\/li>\n \t<li>\\(\\left(6,6\\right)\\)<\/li>\n \t<li>\\(\\left(5,0\\right)\\)<\/li>\n \t<li>\\(\\left(-2,7\\right)\\)<\/li>\n \t<li>\\(\\left(-5,2\\right)\\)<\/li>\n \t<li>\\(\\left(-1,7\\right)\\)<\/li>\n \t<li>\\(\\left(-3,5\\right)\\)<\/li>\n \t<li>(10, 12)<\/li>\n \t<li>\\(\\left(\\frac{1}{2},3\\right)\\)<\/li>\n \t<li>\\(\\left(\\frac{1}{2},-\\frac{3}{4}\\right)\\)<\/li>\n \t<li>Infinitely many solutions<\/li>\n \t<li>Infinitely many solutions<\/li>\n \t<li>No solution<\/li>\n \t<li>No solution<\/li>\n \t<li>The numbers are 6 and 9.<\/li>\n \t<li>The numbers are \u22127 and \u221219.<\/li>\n \t<li>The length is 20 and the width is 10.<\/li>\n \t<li>The length is 34 and the width is 8.<\/li>\n \t<li>The measures are 16\u00b0 and 74\u00b0.<\/li>\n \t<li>The measures are 45\u00b0 and 45\u00b0.<\/li>\n \t<li>80 cable packages would need to be sold.<\/li>\n \t<li>Mitchell would need to sell 120 stoves.<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n \t<li>\\(t=2\\) hours<\/li>\n \t<li>\\(s=2\\frac{1}{2}\\) hours<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\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>Solve a system of equations by substitution<\/li>\n<li>Solve applications of systems of equations by substitution<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345194538\">Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> both between \u221210 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.<\/p>\n<p id=\"fs-id1168345297967\">In this section, we will solve systems of linear equations by the substitution method.<\/p>\n<div id=\"fs-id1168345216137\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Equations by Substitution<\/h1>\n<p id=\"fs-id1168345301534\">We will use the same system we used first for graphing.<\/p>\n<div id=\"fs-id1168345415079\" 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-id1168345249949\">We will first solve one of the equations for either <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>. We can choose either equation and solve for either variable\u2014but we\u2019ll try to make a choice that will keep the work easy.<\/p>\n<p id=\"fs-id1168345644836\">Then we substitute that expression into the other equation. The result is an equation with just one variable\u2014and we know how to solve those!<\/p>\n<p id=\"fs-id1168345416782\">After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How to Solve a System of Equations by Substitution<\/div>\n<div id=\"fs-id1168341959560\" data-type=\"exercise\">\n<div id=\"fs-id1168345656724\" data-type=\"problem\">\n<p id=\"fs-id1168345724784\">Solve the system by substitution. <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;\" \/><\/p>\n<p id=\"fs-id1171792480266\">\n<\/div>\n<div id=\"fs-id1168345240310\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1168345291516\" data-type=\"media\" data-alt=\"This figure has three columns and six rows. The first row says, \u201cStep 1. Solve one of the equations for either variable.\u201d To the right of this, the middl row reads, \u201cWe\u2019ll solve the first equation for y.\u201d The third column shows the two equations: 2x + y = 7 and x \u2013 2y = 6. It shows that 2x + y = 7 becomes y = 7 \u2013 2x.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_02_001a_img_new.jpg\" alt=\"This figure has three columns and six rows. The first row says, \u201cStep 1. Solve one of the equations for either variable.\u201d To the right of this, the middl row reads, \u201cWe\u2019ll solve the first equation for y.\u201d The third column shows the two equations: 2x + y = 7 and x \u2013 2y = 6. It shows that 2x + y = 7 becomes y = 7 \u2013 2x.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345646781\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2. Substitute the expression from Step 1 into the other equation.\u201d Then, \u201cWe replace y in the second equation with the expression 7 \u2013 2x.\u201d It then shows the x \u2013 2y = 6 becomes x \u2013 2(7 \u2013 2x) = 6.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2. Substitute the expression from Step 1 into the other equation.\u201d Then, \u201cWe replace y in the second equation with the expression 7 \u2013 2x.\u201d It then shows the x \u2013 2y = 6 becomes x \u2013 2(7 \u2013 2x) = 6.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345291590\" data-type=\"media\" data-alt=\"The third row says, \u201cStep 3: Solve the resulting equation.\u201d Then \u201cNow we have an equation with just 1 variable. WE know how to solve this!\u201d It then shows that x \u2013 2(7 \u2013 2x) = 6 becomes x \u2013 14 + 4x = 6 which becomes 5x = 20. Thus x = 4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001c_img_new.jpg\" alt=\"The third row says, \u201cStep 3: Solve the resulting equation.\u201d Then \u201cNow we have an equation with just 1 variable. WE know how to solve this!\u201d It then shows that x \u2013 2(7 \u2013 2x) = 6 becomes x \u2013 14 + 4x = 6 which becomes 5x = 20. Thus x = 4.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345328036\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4. Substitute the solution in Step 3 into one of the original quaitons to find the other variable.\u201d Then, \u201cWe\u2019ll use the first equation and replace x with 4.\u201d Then it shows that 2x + y = 7 becomes 2(4) + y = 7. This becomes 8 + y = 7, and thus y = \u22121.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4. Substitute the solution in Step 3 into one of the original quaitons to find the other variable.\u201d Then, \u201cWe\u2019ll use the first equation and replace x with 4.\u201d Then it shows that 2x + y = 7 becomes 2(4) + y = 7. This becomes 8 + y = 7, and thus y = \u22121.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345254426\" data-type=\"media\" data-alt=\"The fifth row reads, \u201cStep 5. Write the solution as an ordered pair.\u201d Then \u201cThe ordered air is (x, y).\u201d Then (4, \u22121).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001e_img_new.jpg\" alt=\"The fifth row reads, \u201cStep 5. Write the solution as an ordered pair.\u201d Then \u201cThe ordered air is (x, y).\u201d Then (4, \u22121).\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345287958\" data-type=\"media\" data-alt=\"The sixth row reads, \u201cStep 6. Check that the order pair is a solution to both original equations.\u201d Then, \u201cSubstitute (4, \u22121) into both equations and make sure they are both true.\u201d It then shows that 2x + y = 7 becomxe 2(4) + \u22121 = 7, and thus 7 = 7. It also shows that x \u2013 2y = 6 becomes 4 \u2013 2(\u22121) = 6, and thus 6\u22126. It also states, \u201cBoth equations are ture. (4, \u22121) is the solution to the system.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_001f_img_new.jpg\" alt=\"The sixth row reads, \u201cStep 6. Check that the order pair is a solution to both original equations.\u201d Then, \u201cSubstitute (4, \u22121) into both equations and make sure they are both true.\u201d It then shows that 2x + y = 7 becomxe 2(4) + \u22121 = 7, and thus 7 = 7. It also shows that x \u2013 2y = 6 becomes 4 \u2013 2(\u22121) = 6, and thus 6\u22126. It also states, \u201cBoth equations are ture. (4, \u22121) is the solution to the system.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341852430\" data-type=\"problem\">\n<p id=\"fs-id1168345417415\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-938d7c761876ab0affaf7d41f10dc0ee_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;&#121;&#61;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#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=\"138\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168341961510\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345408042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc9db18ceda8b325515059e9c425b44f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How to solve a system of equations by substitution.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169753231587\" class=\"stepwise\" type=\"1\">\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 1 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute the solution in Step 3 into one of the original equations to find the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345216137\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345340004\">If one of the equations in the system is given in slope\u2013intercept form, Step 1 is already done!<\/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 id=\"fs-id1168341960594\" data-type=\"problem\">\n<p id=\"fs-id1168345203647\">Solve the system by substitution.<\/p>\n<p id=\"fs-id1169751032452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f054c417dcea938e603530d885b91de1_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#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=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345284532\" 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-id1168345254082\">The second equation is already solved for <em data-effect=\"italics\">y<\/em>. We will substitute the expression in place of <em data-effect=\"italics\">y<\/em> in the first equation.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167836448172\" style=\"width: 100%;\" summary=\"In this figure we have two columns that work out the solution. In the top of the right column is the system of equation \u201cx plus y equals negative 1\u201d and \u201cy equals x plus 5.\u201d The next line down on the left reads, \u201cThe second equation is already solved for y. We will substitute into the first equation. Next to this is \u201cy equals x plus 5. The \u201cx plus 5\u201d is within a red circle with an arrow pointing to the \u201cy\u201d in the next line, which indicates that \u201cx plus 5\u201d will be the substitute for \u201cy\u201d. The next line on the left reads, \u201cReplace the \u201cy\u201d with \u201cx plus 5\u201d. Then, \u201cSolve the resulting equation for \u201cx\u201d. Next to this on the right is the equation \u201cx plus x plus 5 equals negative 1.\u201d Under this is is \u201c2x plus 5 equals negative 1\u201d, under this is\u201d2x equals negative 6\u201d and under this is \u201cx equals negative 3\u201d. The \u201cequals negative 3\u201d has a red circle around it with an arrow pointing to the \u201cx\u201d in the expression below. The next line down on the left reads, \u201cSubstitute x plus negative 3 into \u201cy equals x plus 5 to find y\u201d. To the right of this is the expression \u201cy equals x plus 5\u201d. Under this is \u201cy equals negative 3 plus 5\u201d, and under this is \u201cy equals 2\u201d. The next line down on the left reads, \u201cthe ordered pair is negative 3, 2\u201d in parentheses. Below this it reads, \u201cCheck the ordred pair in both equations.\u201d Below this we have both sets of equations. On the left if x plus y equals negative 1, under this negative 3 plus 2 equals negative one. A question mark is above the equal sign. Under this is negative 1 equals negative 1 with a checkmark beside it. The equation to the right of this reads, y equals x plus 5. Undr this is 2 equals negative 3 plus 5 with a questions mark above the equal sign. Under this is 2 equals 2, with a checkmark beside it. In the right column, the last line reads, \u201cthe solution is ordered pair negative 3 and 2.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836700889\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">The second equation is already solved for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/>\n<\/span>We will substitute into the first equation.<\/td>\n<td style=\"width: 295px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"bottom\">Replace the <em data-effect=\"italics\">y<\/em> with <em data-effect=\"italics\">x<\/em> + 5.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836700881\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836329838\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836575027\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836605298\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"bottom\">Substitute <em data-effect=\"italics\">x<\/em> = \u22123 into <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">x<\/em> + 5 to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836492201\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836534698\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">The ordered pair is (\u22123, 2).<\/td>\n<td style=\"width: 295px;\" data-valign=\"top\"><span id=\"fs-id1167836618806\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_002h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-08aa2862c5fa58e4cfb668483bd52d4a_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;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#51;&#43;&#53;&#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;&#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=\"64\" width=\"297\" style=\"vertical-align: -26px;\" \/><\/td>\n<td style=\"width: 295px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 555px;\"><\/td>\n<td style=\"width: 295px;\" data-align=\"center\" data-valign=\"top\">The solution is (\u22123, 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 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345448675\" data-type=\"problem\">\n<p id=\"fs-id1168345374070\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-732901cf2b96cc28c4906a5b59305c5f_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;&#121;&#61;&#51;&#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=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345541402\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345254372\"><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;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345451707\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345557251\" data-type=\"exercise\">\n<div id=\"fs-id1168345448675\" data-type=\"problem\">\n<p><span style=\"text-align: initial; font-size: 14pt;\">If the equations are given in standard form, we\u2019ll need to start by solving for one of the variables. In this next example, we\u2019ll solve the first equation for <\/span><em style=\"text-align: initial; font-size: 14pt;\" data-effect=\"italics\">y.<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\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-id1168345746371\" data-type=\"problem\">\n<p id=\"fs-id1168345452943\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-db994b8f21679981a890941db48b8be8_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;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#52;&#121;&#61;&#45;&#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=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345415462\" 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-id1168345285867\">We need to solve one equation for one variable. Then we will substitute that expression into the other equation.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167829753382\" style=\"width: 100%;\" summary=\"This figure show two equations: 3x + y = 5 and 2x + 4y = -10. It indicates, \u201cSolve for y.\u201d and shows 3x + y = 5 becomes y = -3x + 5. Then it says, \u201cSubstitute into the other equation. Relace the y with -3x + 5. Solve the resulting equation for x.\u201d It then shows that 2x + 4y = -10 becomes 2x + 4(-3x +5) = -10. This then becomes 2x -12x +20 = -10 which becomes -10x + 20 = -10. This becomes -10x = -30 which means x = 3. The figure then says, \u201cSubstitute x = 3 into 3x + y = 5 to find y.\u201d Thus 3x + y = 5 becomes 3(3) + y = 5. This becomes 9 + y = 5 which means y = -4. The figure then indicates, \u201cThe ordered pair is (3, -4). The figure then instructs, \u201cCheck the ordered pari in both equations:\u201d Thus 3x + y = 5 becomes 3 times 3 + (-4) = 5. This is 9 \u2013 4 = 5 which is 5 = 5. 2x + 4y = -10 becomes 2 times 3 + 4(-4) = -10 which means 6 \u2013 16 = -10 and thus -10 = -10. The figure then says, \u201cThe solution is (3, -4).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"bottom\">Solve for <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute into the other equation.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836318910\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Replace the <em data-effect=\"italics\">y<\/em> with \u22123<em data-effect=\"italics\">x<\/em> + 5.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167832994444\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836544330\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829719558\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829893884\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"bottom\">Substitute <em data-effect=\"italics\">x<\/em> = 3 into 3<em data-effect=\"italics\">x<\/em> + <em data-effect=\"italics\">y<\/em> = 5 to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167836556269\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\"><\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829908075\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167833136578\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">The ordered pair is (3, \u22124).<\/td>\n<td style=\"width: 46.5347%;\" data-valign=\"top\"><span id=\"fs-id1167829696105\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_003i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-027751eda629e7d7cca7e5c358bcfe1c_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;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#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;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#45;&#52;&#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;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#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;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#45;&#49;&#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;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#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=\"89\" width=\"391\" style=\"vertical-align: -38px;\" \/><\/td>\n<td style=\"width: 46.5347%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.2673%;\"><\/td>\n<td style=\"width: 46.5347%;\" data-align=\"center\" data-valign=\"top\">The solution is (3, \u22124).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345746702\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1168345426936\" data-type=\"exercise\">\n<div id=\"fs-id1168345384384\" data-type=\"problem\">\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-id1168345384384\" data-type=\"problem\">\n<p id=\"fs-id1168341864330\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b2b70faf920b5b02f2455e5cbbbec765_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;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#50;&#121;&#61;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345250252\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345213782\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a685e6a8556e8135bece1243dc70fe1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345386232\">In example 3, it was easiest to solve for <em data-effect=\"italics\">y<\/em> in the first equation because it had a coefficient of 1. In the next example it will be easier to solve for <em data-effect=\"italics\">x<\/em>.<\/p>\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-id1168345436528\" data-type=\"problem\">\n<p id=\"fs-id1168341840994\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-baba69bd2fcbe1f4a9b0750093fd8da8_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;&#50;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#50;&#121;&#61;&#51;&#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-id1168345256708\" 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-id1168345253030\">We will solve the first equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and then substitute the expression into the second equation.<\/p>\n<table id=\"fs-id1167829596505\" style=\"width: 100%;\" summary=\"This figure says, \u201cSolve for x.\u201d Thus x \u2013 27 = -2 becomes x = 2y -2. The figure then instructs, \u201cSubstitute into the other equations Replace x with 2y -2. Solve for the resulting equation for y.\u201d Thus 3x + 2y = 34 becomes 3(2y -2) +2y = 34. This then becomes 6y \u2013 6 +2y = 34. This becomes 8y \u2013 6 = 34 or 8y = 40. Thus y = 5. The figure then instructs, \u201cSubstitute y = 5 inot x \u2013 2y = -2 to find x.\u201d Thus x \u2013 2y = -2 becomes x \u2013 2 times 5 = -2. This becomes x \u2013 10 = -2 which menas x = 8. The figure then instructs, \u201cThe ordered pair is (8, 5). Check the ordered pair in both equations:\u201d It then shows that x \u2013 2y = -2 becomes 8 \u2013 2 times 5 = -2. This becomes 8 \u2013 10 = -2 and thus -2 = -2. This also shows that 3x + 2y = 34 becomes 3 times 8 plus 2 times 5 = 34. This becomes 24 + 10 = 34 and thus 34 = 34. The figure then states, \u201cThe solution is (8, 5).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 297.062px;\"><\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167835384764\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Solve for <em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute into the other equation.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167836361817\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Replace the <em data-effect=\"italics\">x<\/em> with 2<em data-effect=\"italics\">y<\/em> \u2212 2.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167833031496\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Solve the resulting equation for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167836287768\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"middle\"><span data-type=\"newline\"><br \/>\n<\/span>Substitute <em data-effect=\"italics\">y<\/em> = 5 into <em data-effect=\"italics\">x<\/em> \u2212 2<em data-effect=\"italics\">y<\/em> = \u22122 to find <em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 178.156px;\" data-valign=\"top\"><span id=\"fs-id1167833136523\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167832945647\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836503161\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829691575\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836521554\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836388843\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_004j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">The ordered pair is (8, 5).<\/td>\n<td style=\"width: 178.156px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e38abefcea2ca39fac96ae7f09ddf559_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;&#50;&#121;&#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;&#56;&#45;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#53;&#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;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#45;&#49;&#48;&#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;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#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;&#51;&#120;&#43;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#56;&#43;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#53;&#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;&#51;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#52;&#43;&#49;&#48;&#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;&#51;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#52;&#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=\"88\" width=\"345\" style=\"vertical-align: -38px;\" \/><\/td>\n<td style=\"width: 178.156px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 297.062px;\"><\/td>\n<td style=\"width: 178.156px;\" data-align=\"center\" data-valign=\"top\">The solution is (8, 5).<\/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-id1168345302419\" data-type=\"problem\">\n<p id=\"fs-id1168345452325\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e50df861a5e3e4e6d63e8e19bd83182_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;&#53;&#121;&#61;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#51;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345231497\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345346380\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e6b25c6279fe53efe7c78796a156eeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345301408\">When both equations are already solved for the same variable, it is easy to substitute!<\/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-id1168345541944\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345230089\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3dc3ba5b41d14d97a851102380a5156d_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;&#53;&#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=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345742979\" style=\"font-size: 16.8px;\" 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-id1168345453561\">Since both equations are solved for\u00a0<em data-effect=\"italics\">y<\/em>, we can substitute one into the other.<\/p>\n<table id=\"fs-id1167829714127\" class=\"unnumbered unstyled can-break\" summary=\"This figure instructs, \u201cSubstitute (1\/2)x for y in the first equation. Replace the y with (1\/2)x. Solve the resulting equation. Start by clearing the fraction. Solve for x.\u201d Thus y = (1\/2)x substituts into y = -2x +5. This becomes (1\/2)x = -2x +5. Then 2((1\/2)x) = 2(-2x + 5) which becomes x = -4x +10. This becomes 5x =10 which means x = 2. The figure then instructs, \u201cSubstitute x = 2 into y = (1\/2)x to find y. Thus y = (1\/2)x becomes y = (1\/2) times 2, and thus y = 1. The figure then says, \u201cThe ordered pair is (2, 1). It then instructs, \u201cCheck the ordered pair in both equations:\u201d Thus y = (1\/2)x becomes 1 = 1\/2 times 2, and thus 1 = 1. Y = -2x +5 becomes 1 = -2 times 2 + 5. This becomes 1 = -4 +5 and thus 1 = 1. The figure states, \u201cThe solution is (2, 1).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"bottom\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b61981392063277276609ee2214ce59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"19\" style=\"vertical-align: -6px;\" \/> for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167829746913\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2057f20433f2f36767f902adebdc5ca9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"23\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836329209\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Solve the resulting equation. Start<span data-type=\"newline\"><br \/>\n<\/span>by clearing the fraction.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836548764\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167833086359\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\"><\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167836619846\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\">Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 2 into\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b61981392063277276609ee2214ce59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"19\" style=\"vertical-align: -6px;\" \/> to find\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 262.531px;\" data-valign=\"top\"><span id=\"fs-id1167824735799\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167824732580\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836352956\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_005h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">The ordered pair is (2,1).<\/td>\n<td style=\"width: 262.531px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\" data-valign=\"top\">Check the ordered pair in both equations:<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2d22ead14adbbcb25c95653486062539_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#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;&#50;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"282\" style=\"vertical-align: -36px;\" \/><\/td>\n<td style=\"width: 262.531px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.688px;\"><\/td>\n<td style=\"width: 262.531px;\" data-align=\"center\" data-valign=\"top\">The solution is (2,1).<\/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-id1168345667581\" data-type=\"problem\">\n<p id=\"fs-id1168345216253\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b5d7c9d80b86ae5213208519594627d_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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345256614\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345199012\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ccc35b92a6567ced556cb46473589564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#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-id1168341961701\">Be very careful with the signs in the next example.<\/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-id1168345293219\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345727316\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-70b115a2ba1d94617b572a15c4bd3b3b_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;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345424365\" style=\"font-size: 16.8px;\" 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-id1168345300933\">We need to solve one equation for one variable. We will solve the first equation for\u00a0<em data-effect=\"italics\">y<\/em>.<\/p>\n<table id=\"fs-id1167836625241\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSolve the first equation for y.\u201d Thus, 4x + 2y = 4 becomes 2y = -4x + 4. This then becomes y = -2x + 2. The figure then instructs, \u201cSubstitute -2x + 2 for y in the second equation. Replace the y with -2x +2. Solve the equation for x.\u201d Thus 6x \u2013 y = 8 becomes 6x \u2013 (-2x + 2) = 8. This becomes 6x + 2x -2) = 8. This becomes 8x \u2013 2 = 8. Thus 8x = 10 and x = 5\/4. The figure then instructs, \u201cSubstitute x = 5\/4 into 4x +2y = 4 to find y. Thus 4x +2y = 4 becomes 4(5\/4) + 2y = 4. This becomes 5 + 2y = 4 or 2y = -1, and thus y = -1\/2. The figure then says, \u201cThe ordered pair is (5\/4, -1\/2).\u201d It then instructs, \u201cCheck the ordered pair in both equations:\u201d Thus 4x +2y = 4 becomes 4(5\/4) + 2(-1\/2) = 4. This becomes 5 \u2013 1 = 4 and thus 4 = 4. 6x \u2013 y = 8 becomes 6(5\/4) \u2013 (-1\/2) = 8. This becomes 15\/4 \u2013(-1\/2) = 8. Thus 16\/2 = 8 and 8 = 8. The figure then says, \u201cThe solution is (5\/4, -1\/2).\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836628457\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Solve the first equation for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836368137\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"bottom\">Substitute \u22122<em data-effect=\"italics\">x<\/em>\u00a0+ 2 for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the second equation.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836698748\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with \u22122<em data-effect=\"italics\">x<\/em>\u00a0+ 2.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"bottom\"><span id=\"fs-id1167836388313\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Solve the equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"bottom\"><span id=\"fs-id1167836359959\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836299914\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836568022\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\"><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-890be6c8bbcdea64df661b746e7455d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> into 4<em data-effect=\"italics\">x<\/em>\u00a0+ 2<em data-effect=\"italics\">y<\/em>\u00a0= 4 to find\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 45.9406%;\" data-valign=\"top\"><span id=\"fs-id1167836549263\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829693698\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836310171\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829596793\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836315024\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_006l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">The ordered pair is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d7d76bb674d165b6af23c5553f6a9f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 45.9406%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\" data-valign=\"top\">Check the ordered pair in both equations.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-450c037d11f81b295cf3c5e65425c2aa_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;&#52;&#120;&#43;&#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;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#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;&#53;&#45;&#49;&#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;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#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;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#54;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#52;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#50;&#125;&#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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"146\" width=\"393\" style=\"vertical-align: -52px;\" \/><\/td>\n<td style=\"width: 45.9406%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 53.8614%;\"><\/td>\n<td style=\"width: 45.9406%;\" data-align=\"center\" data-valign=\"top\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d7d76bb674d165b6af23c5553f6a9f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/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-id1168345560310\" data-type=\"problem\">\n<p id=\"fs-id1168345436701\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9d868b4bd4b0830531bd547f59f00e34_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;&#52;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#52;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345416943\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345370970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e8fb98e4e344485819d20845a65ffcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345744719\">In the next example, it will take a little more work to solve one equation for <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>.<\/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 style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345290615\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de0a626d967e0e4593c05c0bdb8e1322_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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#53;&#121;&#45;&#50;&#48;&#120;&#61;&#45;&#51;&#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=\"152\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345285072\" style=\"font-size: 16.8px;\" 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-id1168345424353\">We need to solve one equation for one variable. We will solve the first equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/p>\n<table id=\"fs-id1167829692301\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSolve the first equation for x.\u201d Thus 4x -3y = 6 becomes 4x = 3y + 6. This becomes x = (3\/4)y + 3\/2. The figure then instructs, \u201cSubstitute (3\/4)y + 3\/2 for x in the section equation. Replace the x with (3\/4)y + 3\/2. Solve for y.\u201d Thus 15y -20x = -30 becomes 15y \u2013 20((3\/4)y + 3\/2) = -30. Then 15y -15y -30 = -30 becomes 0 -30 = -30 or 0 = 0.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167825998321\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"top\">Solve the first equation for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836665073\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"bottom\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9e1b21a34d89dafc40eba05c0cda396c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/> for\u00a0<em data-effect=\"italics\">x<\/em>\u00a0in the second equation.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836352297\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">x<\/em>\u00a0with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-918ce581cb9053710060157f646606b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836549360\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\" data-valign=\"bottom\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836731170\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167829859295\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.703%;\"><\/td>\n<td style=\"width: 50.099%;\" data-valign=\"top\"><span id=\"fs-id1167836330142\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_007g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345242251\">Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.<\/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 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345408367\" data-type=\"problem\">\n<p id=\"fs-id1168345373805\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ae8ff376731c51fadb7598163ac23839_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;&#51;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#49;&#50;&#121;&#43;&#56;&#120;&#61;&#52;&#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=\"143\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345439980\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345196602\">infinitely many solutions<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341923833\">Look back at the equations in example 6. Is there any way to recognize that they are the same line?<\/p>\n<p id=\"fs-id1168345544175\">Let\u2019s see what happens in the next example.<\/p>\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-id1168345473672\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168341955763\">Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9e5af263db3b0f7ed038c3e957433234_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;&#53;&#120;&#45;&#50;&#121;&#61;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1168345695456\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1168345273788\">The second equation is already solved for\u00a0<em data-effect=\"italics\">y<\/em>, so we can substitute for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/p>\n<table id=\"fs-id1167836635488\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cSubstitue f for y in the first equation. Replace the y with (5\/2)x. Solve for x.\u201d Thus y = (5\/2)x is substituted is not 5x minus 2((5\/2)x) = negative 10. This becomes 5x \u2013 5x = negative 10 or 0 = negative 10.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0for\u00a0<em data-effect=\"italics\">y<\/em>\u00a0in the first equation.<\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"bottom\"><span id=\"fs-id1167836704941\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"top\">Replace the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5d2c1d48d3747b4dfcc30667b21335c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"23\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\"><span id=\"fs-id1167836331212\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\" data-valign=\"bottom\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\"><span id=\"fs-id1167836361374\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.0792%;\"><\/td>\n<td style=\"width: 47.7228%;\" data-valign=\"top\" data-align=\"right\"><span id=\"fs-id1167836323046\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_008d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168341863633\">Since 0 = \u221210 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve the system by substitution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d110aeb914f5c985957c3b214a255e68_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;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#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=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<div id=\"fs-id1168345210433\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345210581\">\u00a0 \u00a0no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341864163\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Substitution<\/h1>\n<p id=\"fs-id1168345373909\">We\u2019ll copy here the problem solving strategy we used in the last sub chapter for solving systems of equations. Now that we know how to solve systems by substitution, that\u2019s what we\u2019ll do in Step 5.<\/p>\n<div id=\"fs-id1168345574727\" 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<ol id=\"fs-id1169746619671\" 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<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341923605\">Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example\u2014how would you have done it with just one variable?<\/p>\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-id1168345414503\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345414731\">The sum of two numbers is zero. One number is nine less than the other. Find the numbers.<\/p>\n<\/div>\n<div id=\"fs-id1168345744922\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836529197\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two numbers. Step 3. Name what we are looking for. Let n = the first number and let m = the second number. Step 4. Translate into a system of equations. The sum of two numbers is zero. N + m = 0. One number is nine less than the other or n = m -9. The system is: n + m = 0 and n = m \u2013 9. Step 5. Solve the system of equations. We will use substitution since the second equation is solved for ns. Substitute m -9 for n in the first equations. Solve for m.\u201d Thus n = m \u2013 9 is substituted into n + m = 0 and becomes m \u2013 9 + m = 0 or 2m \u2013 9 = 0. Thus 2m = 9 and m = 9\/2. The figure then instructs, \u201cSubstitute m = 9\/2 into the second equation and then solve for n.\u201d Thus n = m \u2013 9 bexomes m = 9\/2 \u2013 9 or m = 9\/2 \u2013 18\/2. Thus n = -9\/2. The figure then instructs, \u201cStep 6. Check the answer in the problem. Does these numbers make sense in the problem? We will leave this to you!\u201d Step 7. Answer the question. The numbers are 9\/2 and \u2013 9\/2.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">We are looking for two numbers.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">\u00a0Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-602c4d5dfa06ccbd52974998a84a8f4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"29\" style=\"vertical-align: 0px;\" \/> the first number<span data-type=\"newline\"><br \/>\n<\/span>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0f5541d857afef6f1c8562acca75ad12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"33\" style=\"vertical-align: 0px;\" \/> the second number<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The sum of two numbers is zero.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829844007\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">One number is nine less than the other.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836613186\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836551442\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of<span data-type=\"newline\"><br \/>\n<\/span>equations. We will use substitution<span data-type=\"newline\"><br \/>\n<\/span>since the second equation is solved<span data-type=\"newline\"><br \/>\n<\/span>for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">m<\/em>\u00a0\u2212 9 for\u00a0<em data-effect=\"italics\">n<\/em>\u00a0in the first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836415079\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">m<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836294980\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009f_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-id1167833139030\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009g_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-id1167836327185\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-16bd110d9406c389f9c9a98bdd3ae2df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> into the second equation<span data-type=\"newline\"><br \/>\n<\/span>and then solve for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"bottom\"><span id=\"fs-id1167833058872\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009i_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-id1167836555974\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009j_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-id1167836688249\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009k_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-id1167833139392\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_009l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">Do these numbers make sense in<span data-type=\"newline\"><br \/>\n<\/span>the problem? We will leave this to you!<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The numbers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-36a0158b653269485337be6797772c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0460e95f6f820261fade7dd5882d8605_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/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 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345275137\" data-type=\"problem\">\n<p id=\"fs-id1168345241744\">The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.<\/p>\n<\/div>\n<div id=\"fs-id1168345742616\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345461982\">The numbers are 3 and 7.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168341916379\">In the next example, we\u2019ll use the formula for the perimeter of a rectangle, <em data-effect=\"italics\">P<\/em> = 2<em data-effect=\"italics\">L<\/em> + 2<em data-effect=\"italics\">W<\/em>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345275902\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345560167\">The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.<\/p>\n<\/div>\n<div id=\"fs-id1168345252894\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836686435\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem.\u201d There is a drawing of a rectangle with the width (w) and length (l) marked. The figure instructs, \u201cStep 2. Identify what you are looking for. We are looking for the length and width. Step 3. Name what we are looking for. Let L = the length, W = the width. Step 4. Translate into a system of equations. The perimeter of the rectangle is 88. 2L + 2W = 88. The length is five more than twice the width. L = 2W + 5. The system is 2L + 2W = 88 and L = 2W + 5. Step 5. Solve the system of equations. We will use substation since the second equation is solved of L. Substitute 2W + 5 for L in the first equations. Solve for W.\u201d Thus 2L + 2W = 88 becomes 2(2W + 5) + 2W = 88 which becomes 4W + 10 + 2W = 22. This becomes 6W +10 = 88 or 6W = 78 or W = 13. The figure then instructs, \u201cSubstitute W = 13 into the second equation and then solve for L. L = 2W + 5 becomes L = 2 times 13 + 5. Thus L = 31. The figure then instructs, \u201cStep 6. Check that answer in the problem. Does a rectangle with length 31 and width 18 have perimeter 88? Yes. Step 7. Answer the question. The length is 31 and the width is 13.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829921210\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the length and width.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-549284767161ef0d3728f1f8c7bce365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/> the length<span data-type=\"newline\"><br \/>\n<\/span>\u2003\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-53ae1510ad746496e483726313ec56b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"37\" style=\"vertical-align: 0px;\" \/> the width<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The perimeter of a rectangle is 88.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">\u2003\u2003\u00a0\u00a02<em data-effect=\"italics\">L<\/em>\u00a0+ 2<em data-effect=\"italics\">W<\/em>\u00a0=\u00a0<em data-effect=\"italics\">P<\/em><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836415488\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">The length is five more than twice the width.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836294986\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829743944\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\"><br \/>\n<\/span>We will use substitution since the second<span data-type=\"newline\"><br \/>\n<\/span>equation is solved for\u00a0<em data-effect=\"italics\">L<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute 2<em data-effect=\"italics\">W<\/em>\u00a0+ 5 for\u00a0<em data-effect=\"italics\">L<\/em>\u00a0in the first equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836362612\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">W<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836363048\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010h_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-id1167829739841\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010i_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-id1167836546317\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010j_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-id1167832950892\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"bottom\">Substitute\u00a0<em data-effect=\"italics\">W<\/em>\u00a0= 13 into the second<span data-type=\"newline\"><br \/>\n<\/span>equation and then solve for\u00a0<em data-effect=\"italics\">L<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836558356\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010l_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-id1167833379409\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010m_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-id1167836507472\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_010n_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">Does a rectangle with length 31 and width<span data-type=\"newline\"><br \/>\n<\/span>13 have perimeter 88? Yes.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the equation.<\/td>\n<td data-valign=\"top\">The length is 31 and the width is 13.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341952485\" data-type=\"problem\">\n<p id=\"fs-id1168345274903\">The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.<\/p>\n<\/div>\n<div id=\"fs-id1168345724938\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345297385\">The length is 12 and the width is 8.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345368011\">For the next example we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.<\/p>\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-id1168345745360\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345579068\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.<\/p>\n<\/div>\n<div id=\"fs-id1168345254941\" style=\"font-size: 16.8px;\" 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-id1168345483406\">We will draw and label a figure.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167836616035\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1: Read the problem.\u201d This is a right triangle drawn with one angle labeled a and one angle labeled b. The figure further instructs, \u201cStep 2: Identify what we are looking for. We are looking for measures of the angles. Step 3: Name what we are looking for. Let a = the measure of the first angel and b = the measure of the second angle. Step 4: Translate into a system of equations. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other angle.\u201d Thus a = 3b + 10. \u201cThe sume of the measure of tha angles of a triangle is 180.\u201d Thus, a + b + 90 = 180. \u201cThe system is: a = 3b + 10, a + b + 90 = 180. Step 5. Solve the system of equations. We will use substitution since the first equation is solved for a. Substitute 3b +10 for a in the second equation. Solve for b.\u201d thus a + b + 90 = 180 becomes (3b + 10) + b + 90 = 180. This becomes 4b +100 = 18 or 4b = 80. Thus b = 20. The figure then instructs, \u201cSubstitute b = 20 into the first equation and then solve for a.\u201d Thus a = 3b + 10 becomes a = 3 times 20 +10 or a = 70. The figure then instructs, \u201cStep 6: Check the answer in the problem. We will leave this to you! Step 7: Answer the question. The measures of the small angels are 20 and 70.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836300370\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the measures of the angles.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6fdf9894e43ac86e8b63ba96c72e7181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"27\" style=\"vertical-align: 0px;\" \/> the measure of the 1<sup>st<\/sup>\u00a0angle<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cdf71bbe58eb1a3f8789c2cc199d76dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: 0px;\" \/> the measure of the 2<sup>nd<\/sup>\u00a0angle<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">The measure of one of the small angles<span data-type=\"newline\"><br \/>\n<\/span>of a right triangle is ten more than three<span data-type=\"newline\"><br \/>\n<\/span>times the measure of the other small angle.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829720983\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">The sum of the measures of the angles of<span data-type=\"newline\"><br \/>\n<\/span>a triangle is 180.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836416553\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167829784979\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\"><br \/>\n<\/span>We will use substitution since the first<span data-type=\"newline\"><br \/>\n<\/span>equation is solved for\u00a0<em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836550173\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute 3<em data-effect=\"italics\">b<\/em>\u00a0+ 10 for\u00a0<em data-effect=\"italics\">a<\/em>\u00a0in the<span data-type=\"newline\"><br \/>\n<\/span>second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836684101\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Solve for\u00a0<em data-effect=\"italics\">b<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836523331\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011h_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-id1167829594592\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011i_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-id1167836560598\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute\u00a0<em data-effect=\"italics\">b<\/em>\u00a0= 20 into the first<span data-type=\"newline\"><br \/>\n<\/span>equation and then solve for\u00a0<em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836525449\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011k_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167829711937\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_011l_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer in the problem.<\/td>\n<td data-valign=\"top\">We will leave this to you!<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The measures of the small angles are<span data-type=\"newline\"><br \/>\n<\/span>20 and 70.<\/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 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345287011\" data-type=\"note\">\n<div id=\"fs-id1168345541526\" data-type=\"exercise\">\n<div id=\"fs-id1168345432468\" data-type=\"problem\">The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345250240\" data-type=\"note\">\n<div id=\"fs-id1168345551489\" data-type=\"exercise\">\n<div id=\"fs-id1168345441083\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345324882\">The measure of the angles are 22 degrees and 68 degrees.<\/p>\n<\/details>\n<\/div>\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-id1168345388247\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p>Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her &#36;25,000 plus &#36;15 for each training session. Option B would pay her &#36;10,000 + &#36;40 for each training session. How many training sessions would make the salary options equal?<\/p>\n<\/div>\n<div id=\"fs-id1168345432359\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167836630339\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what you are looking for. We are looking for the number of training sessions that would make the pay equal. Step 3. Name what we are looking for. Let s = Heather\u2019s salary, n = the number of training session. Step 4. Translate into a system of equations. Option A would pay her ?25,000 plus ?15 for each taining session. S = 25000 + 15n; Option would pay her ?10,000 + ?40 for each training session. S = 10,000 + 40n. The system is s = 25,000 +15n, s = 10,000 + 40n. Step 5. Solve the system of equations. We will use sbustitiution. Substitute 25,000 + 15n for s in the second equation. Solve or n.\u201d Thus s = 10,000 +40n becomes 25,000 + 15 n = 10,000 + 40n. This becomes 25,000 = 10,000 + 25n which becomes 15,000 = 25n or n = 600. The figure then instructs, \u201cStep 6. Check the answer. Are 600 training session a year reasonable? Are the two options equal when n = 600? Step 7. Answer the questions. The salary options would be equal for 600 training sessions.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td data-valign=\"top\">We are looking for the number of training sessions<span data-type=\"newline\"><br \/>\n<\/span>that would make the pay equal.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td data-valign=\"top\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-12ac9f5dee0c0d9acdb247ec52211d05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> Heather\u2019s salary.<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee1750ab3f138676ed789405f7cec8c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> the number of training sessions<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations.<\/td>\n<td data-valign=\"top\">Option A would pay her &#36;25,000 plus &#36;15<span data-type=\"newline\"><br \/>\n<\/span>for each training session.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167832999774\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\">Option B would pay her &#36;10,000 + &#36;40<span data-type=\"newline\"><br \/>\n<\/span>for each training session<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836660076\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">The system is:<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836622610\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\"><br \/>\n<\/span>We will use substitution.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836296961\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Substitute 25,000 + 15<em data-effect=\"italics\">n<\/em>\u00a0for\u00a0<em data-effect=\"italics\">s<\/em>\u00a0in the second equation.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167836533652\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167833060219\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012g_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-id1167836623222\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012h_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-id1167836635265\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_02_012i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td data-valign=\"top\">Are 600 training sessions a year reasonable?<span data-type=\"newline\"><br \/>\n<\/span>Are the two options equal when\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= 600?<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td data-valign=\"top\">The salary options would be equal for 600 training sessions.<\/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 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345415652\" data-type=\"problem\">\n<p id=\"fs-id1168345240744\">Geraldine has been offered positions by two insurance companies. The first company pays a salary of &#36;12,000 plus a commission of &#36;100 for each policy sold. The second pays a salary of &#36;20,000 plus a commission of &#36;50 for each policy sold. How many policies would need to be sold to make the total pay the same?<\/p>\n<\/div>\n<div id=\"fs-id1168345386624\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345345938\">There would need to be 160 policies sold to make the total pay the same.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"text-align: initial; font-size: 14pt;\">Access these online resources for additional instruction and practice with solving systems of equations by substitution.<\/span><\/p>\n<div id=\"fs-id1168345432571\" class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1168345294286\" data-display=\"block\">\n<li><a href=\"http:\/\/www.openstax.org\/l\/25SolvingLinear\">Instructional Video-Solve Linear Systems by Substitution<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25Substitution\">Instructional Video-Solve by Substitution<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168341863466\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Solve a system of equations by substitution<\/strong>\n<ol id=\"fs-id1169753334202\" class=\"stepwise\" type=\"1\">\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 1 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute the solution in Step 3 into one of the original equations to find the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to both original equations.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1 data-type=\"title\">4.2 Exercise Set<\/h1>\n<p id=\"fs-id1169750685787\">In the following exercises, solve the systems of equations by substitution.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a720fc661ea9c6c17a689b46d7c0c067_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;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#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-7ed9df8a733f224c3cdc2d8a5b3e1b10_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;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#61;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-510cae26785f1943a25b251618796a82_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;&#53;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#43;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/span><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3ec6240da1b8a3035b716f1f7774194_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;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d21d9d684102e404434819814ffa8eb2_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;&#53;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#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=\"110\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-48b139f04e9604b136b8e499ccd20510_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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/span><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-db316bdb5d4a13b87fc08a69aac75e19_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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3074dfe8bc4f7976cc1bd9322219e24b_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#52;&#120;&#43;&#121;&#61;&#49;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e792b36e28c0b34499438f0f0dc0bb81_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#53;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4048245d8cad85768f817c0a93b3f2b0_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;&#45;&#50;&#121;&#61;&#45;&#49;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dc5071229d9b4477fd38fcdc98244b8c_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#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=\"118\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-111d514f26fbb7302feca5a07deddd20_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;&#56;&#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;&#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=\"104\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-684b3cf69ca7a0c2f1e62b4513de693c_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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#56;&#120;&#45;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4aa08fbd748723ce4a2fba27497ab3c_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;&#49;&#53;&#120;&#43;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#53;&#120;&#43;&#50;&#121;&#61;&#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=\"139\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-22eab6f406c52978a9f43df13c51da72_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#50;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6c88587aec411fb0f54a70d5e3d71dff_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;&#49;&#54;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#45;&#56;&#121;&#61;&#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=\"130\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-20cb3fd40cb331f141abf774081c0c4a_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;&#52;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98c5e14aea7d9f5ac85291dddbde7097_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;&#55;&#125;&#123;&#56;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#55;&#120;&#43;&#56;&#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=\"44\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1169746349628\">In the following exercises, translate to a system of equations and solve.<\/p>\n<ol start=\"19\">\n<li>The sum of two numbers is 15. One number is 3 less than the other. Find the numbers<\/li>\n<li>The sum of two numbers is \u221226. One number is 12 less than the other. Find the numbers.<\/li>\n<li>The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.<\/li>\n<li>The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.<\/li>\n<li>The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/li>\n<li>The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.<\/li>\n<li>Jackie has been offered positions by two cable companies. The first company pays a salary of &#36; 14,000 plus a commission of &#36;100 for each cable package sold. The second pays a salary of &#36;20,000 plus a commission of &#36;25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?<\/li>\n<li>Mitchell currently sells stoves for company A at a salary of &#36;12,000 plus a &#36;150 commission for each stove he sells. Company B offers him a position with a salary of &#36;24,000 plus a &#36;50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?<\/li>\n<li>\n<p id=\"fs-id1168345550140\">Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-19fb87aa497df18bd1d7ca5aaf58819a_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;&#53;&#54;&#115;&#61;&#55;&#48;&#116;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#115;&#61;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/>.<\/p>\n<ol type=\"a\">\n<li>for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> to find out how long it will take Tina to catch up to Stephanie.<\/li>\n<li>what is the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, the number of hours Stephanie will have driven before Tina catches up to her?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1168345729576\" data-type=\"exercise\">\n<div id=\"fs-id1168345729579\" data-type=\"problem\">\n<p id=\"fs-id1168345729581\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; background-color: initial;\">Answers:<\/span><\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#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-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ef24d382319a3ce81f280194edba003a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-5a00a53e3c5401663e307951ae0d74c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#51;&#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-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#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-9c86c96c2d855b3405d15f3e708f920e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#54;&#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-e3e1928d65786877c787a2d401d9e77e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#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-374b47fb7bf10f554c21530f0ecc88e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#55;&#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-3f4c81b0ee204d386d3641c327024dda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#50;&#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-1143c5fd493ebacf95a36bd9c1eafc1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#55;&#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-5aa29da239e6fed61429835a4b4444af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>(10, 12)<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-db35b1b5fc11e032383ab900b5c4cb7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4524fa7b31d488981d52bd6238506836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#45;&#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=\"55\" style=\"vertical-align: -7px;\" \/><\/li>\n<li>Infinitely many solutions<\/li>\n<li>Infinitely many solutions<\/li>\n<li>No solution<\/li>\n<li>No solution<\/li>\n<li>The numbers are 6 and 9.<\/li>\n<li>The numbers are \u22127 and \u221219.<\/li>\n<li>The length is 20 and the width is 10.<\/li>\n<li>The length is 34 and the width is 8.<\/li>\n<li>The measures are 16\u00b0 and 74\u00b0.<\/li>\n<li>The measures are 45\u00b0 and 45\u00b0.<\/li>\n<li>80 cable packages would need to be sold.<\/li>\n<li>Mitchell would need to sell 120 stoves.<\/li>\n<li>\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eec54adef7198c4decb9169a2b472009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"38\" style=\"vertical-align: 0px;\" \/> hours<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-93d89f48fca4e1e172b6e183f37e758f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/> hours<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":125,"menu_order":2,"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-1244","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\/1244","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\/1244\/revisions"}],"predecessor-version":[{"id":1245,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1244\/revisions\/1245"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/1094"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1244\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1244"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1244"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1244"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}