{"id":1297,"date":"2019-07-29T23:24:57","date_gmt":"2019-07-29T23:24:57","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-elimination\/"},"modified":"2021-08-31T21:20:56","modified_gmt":"2021-08-31T21:20:56","slug":"solve-systems-of-equations-by-elimination","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/solve-systems-of-equations-by-elimination\/","title":{"raw":"4.3  Solve Systems of Equations by Elimination","rendered":"4.3  Solve Systems of Equations by Elimination"},"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 elimination<\/li>\n \t<li>Solve applications of systems of equations by elimination<\/li>\n \t<li>Choose the most convenient method to solve a system of linear equations<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345297121\" class=\"be-prepared\" data-type=\"note\"><\/div>\n<p id=\"fs-id1168345215927\">We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.<\/p>\n<p id=\"fs-id1168345270702\">The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we\u2019ll do with the elimination method, too, but we\u2019ll have a different way to get there.<\/p>\n\n<div id=\"fs-id1168345742458\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Equations by Elimination<\/h1>\n<p id=\"fs-id1168341917939\">The <span class=\"no-emphasis\" data-type=\"term\">Elimination Method<\/span> is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.<\/p>\n<p id=\"fs-id1168345415661\">For any expressions <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em>,<\/p>\n\n<div id=\"fs-id1168345260014\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccc}\\text{if}\\hfill &amp; \\hfill a&amp; =\\hfill &amp; b\\hfill \\\\ \\text{and}\\hfill &amp; \\hfill c&amp; =\\hfill &amp; d\\hfill \\\\ \\text{then}\\hfill &amp; \\hfill a+c&amp; =\\hfill &amp; b+d\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345342752\">To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.<\/p>\n<p id=\"fs-id1168345237002\">Notice how that works when we add these two equations together:<\/p>\n\n<div id=\"fs-id1168345213321\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccccc}3x+y=5\\hfill \\\\ \\text{}{2x-y=0}\\hfill \\\\ 5x\\phantom{\\rule{1.7em}{0ex}}=5\\hfill \\end{array}\\)<\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><span style=\"font-size: 14pt; text-align: initial;\">The <\/span><em style=\"font-size: 14pt; text-align: initial;\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt; text-align: initial;\">\u2019s add to zero and we have one equation with one variable.<\/span><\/div>\n<\/div>\n<div id=\"fs-id1168345742458\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345261092\">Let\u2019s try another one:<\/p>\n\n<div id=\"fs-id1168345418800\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}x+4y=2\\hfill \\\\ 2x+5y=-2\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345376612\">This time we don\u2019t see a variable that can be immediately eliminated if we add the equations.<\/p>\n<p id=\"fs-id1168345558535\">But if we multiply the first equation by \u22122, we will make the coefficients of <em data-effect=\"italics\">x<\/em> opposites. We must multiply every term on both sides of the equation by \u22122.<\/p>\n<span data-type=\"media\" data-alt=\"This figure shows two equations. The first is negative 2 times x plus 4y in parentheses equals negative 2 times 2. The second is 2x + 5y = negative 2. This figure shows two equations. The first is negative 2x minus 8y = negative 4. The second is 2x + 5y = -negative 2.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_03_002_img_new.jpg\" alt=\"This figure shows two equations. The first is negative 2 times x plus 4y in parentheses equals negative 2 times 2. The second is 2x + 5y = negative 2. This figure shows two equations. The first is negative 2x minus 8y = negative 4. The second is 2x + 5y = -negative 2.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168341959314\">Now we see that the coefficients of the <em data-effect=\"italics\">x<\/em> terms are opposites, so <em data-effect=\"italics\">x<\/em> will be eliminated when we add these two equations.<\/p>\n<p id=\"fs-id1168345692963\">Add the equations yourself\u2014the result should be \u22123<em data-effect=\"italics\">y<\/em> = \u22126. And that looks easy to solve, doesn\u2019t it? Here is what it would look like.<\/p>\n<span data-type=\"media\" data-alt=\"This figure shows two equations being added together. The first is negative 2x \u2013 8y = \u22124 and 2x plus 5y = negative 2. The answer is negative 3y = negative 6.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_010_img_new.jpg\" alt=\"This figure shows two equations being added together. The first is negative 2x \u2013 8y = \u22124 and 2x plus 5y = negative 2. The answer is negative 3y = negative 6.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345298294\">We\u2019ll do one more:<\/p>\n\n<div id=\"fs-id1168345262801\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}4x-3y=10\\hfill \\\\ 3x+5y=-7\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345192582\">It doesn\u2019t appear that we can get the coefficients of one variable to be opposites by multiplying one of the equations by a constant, unless we use fractions. So instead, we\u2019ll have to multiply both equations by a constant.<\/p>\nWe can make the coefficients of <em data-effect=\"italics\">x<\/em> be opposites if we multiply the first equation by 3 and the second by \u22124, so we get 12<em data-effect=\"italics\">x<\/em> and \u221212<em data-effect=\"italics\">x<\/em>.\n\n<span id=\"fs-id1168345290693\" data-type=\"media\" data-alt=\"This figure shows two equations. The first is 3 times 4x minus 3y in parentheses equals 3 times 10. The second is negative 4 times 3x plus 5y in parentheses equals negative 4 times negative 7.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_003_img_new.jpg\" alt=\"This figure shows two equations. The first is 3 times 4x minus 3y in parentheses equals 3 times 10. The second is negative 4 times 3x plus 5y in parentheses equals negative 4 times negative 7.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345458655\">This gives us these two new equations:<\/p>\n\n<div id=\"fs-id1168341892603\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{1.1em}{0ex}}12x-9y=30\\hfill \\\\ -12x-20y=28\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168345461401\">When we add these equations,<\/p>\n\n<div id=\"fs-id1168345459263\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccc}\\left\\{\\begin{array}{c}\\phantom{\\rule{1.1em}{0ex}}12x-9y=30\\hfill \\\\ \\text{}{-12x-20y=28}\\hfill \\end{array}\\\\ \\hfill -29y=58\\end{array}\\)<\/div>\n<p id=\"fs-id1169747561781\">the <em data-effect=\"italics\">x<\/em>\u2019s are eliminated and we just have \u221229<em data-effect=\"italics\">y<\/em> = 58.<\/p>\nOnce we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.\n\nNow we\u2019ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.\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 style=\"font-size: 16.8px;\" data-type=\"title\">How to Solve a System of Equations by Elimination<\/div>\n<div id=\"fs-id1168345228558\" style=\"font-size: 16.8px;\" data-type=\"exercise\">\n<div id=\"fs-id1168345427214\" data-type=\"problem\">\n<p id=\"fs-id1168345376536\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168341852589\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<span id=\"fs-id1168341955857\" data-type=\"media\" data-alt=\"This figure has seven rows and three columns. The first row reads, \u201cStep 1. Write both equations in standard form. If any coefficients are fractions, clear them.\u201d It also says, \u201cBoth equations are in standard form, A x + B y = C. There are no fractions.\u201d It also gives the two equations as 2x + y = 7 and x \u2013 2y = 6.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004a_img_new.jpg\" alt=\"This figure has seven rows and three columns. The first row reads, \u201cStep 1. Write both equations in standard form. If any coefficients are fractions, clear them.\u201d It also says, \u201cBoth equations are in standard form, A x + B y = C. There are no fractions.\u201d It also gives the two equations as 2x + y = 7 and x \u2013 2y = 6.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345199327\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2: Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.\u201d It also says, \u201cWe can eliminate the y\u2019s by multiplying the first equation by 2. Multiply both sides of 2x + y = 7 by 2.\u201d It also shows the steps with equations. Initially the equations are ex + y = 7 and x \u2013 2y = 6. Then they become 2(2x + y) = 2 times 7 and x \u2013 2y = 6. They then become 4x + 2y = 14 and x \u2013 2y = 6.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2: Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.\u201d It also says, \u201cWe can eliminate the y\u2019s by multiplying the first equation by 2. Multiply both sides of 2x + y = 7 by 2.\u201d It also shows the steps with equations. Initially the equations are ex + y = 7 and x \u2013 2y = 6. Then they become 2(2x + y) = 2 times 7 and x \u2013 2y = 6. They then become 4x + 2y = 14 and x \u2013 2y = 6.\" data-media-type=\"image\/jpeg\"><\/span><span data-type=\"media\" data-alt=\"The third row says, \u201cStep 3: Add the equations resulting from step 2 to eliminate one variable.\u201d It also says, \u201cWe add the x\u2019s, y\u2019s, and constants.\u201d It then gives the equation as 5x = 20.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004c_img_new.jpg\" alt=\"The third row says, \u201cStep 3: Add the equations resulting from step 2 to eliminate one variable.\u201d It also says, \u201cWe add the x\u2019s, y\u2019s, and constants.\u201d It then gives the equation as 5x = 20.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345511013\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4: Solve for the remaining variable.\u201d It also says, \u201cSolve for x.\u201d It gives the equation as x = 4.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4: Solve for the remaining variable.\u201d It also says, \u201cSolve for x.\u201d It gives the equation as x = 4.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345252882\" data-type=\"media\" data-alt=\"The fifth row says, \u201cStep 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.\u201d It also says, \u201cSubstitute x = 4 into the second equation, x \u2013 2y = 6. Then solve for y.\u201d It then gives the equations as x \u2013 2y = 6 which becomes 4 \u2013 2y = 6. This is then \u22122y = 2, and thus, y = \u22121.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004e_img_new.jpg\" alt=\"The fifth row says, \u201cStep 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.\u201d It also says, \u201cSubstitute x = 4 into the second equation, x \u2013 2y = 6. Then solve for y.\u201d It then gives the equations as x \u2013 2y = 6 which becomes 4 \u2013 2y = 6. This is then \u22122y = 2, and thus, y = \u22121.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345436405\" data-type=\"media\" data-alt=\"The sixth row says, \u201cStep 6: Write the solution as an order pair.\u201d It also says, \u201cWrite it as (x, y).\u201d It gives the ordered pair as (4, \u22121).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004f_img_new.jpg\" alt=\"The sixth row says, \u201cStep 6: Write the solution as an order pair.\u201d It also says, \u201cWrite it as (x, y).\u201d It gives the ordered pair as (4, \u22121).\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345406365\" data-type=\"media\" data-alt=\"The seventh row says, \u201cStep 7: Check that the ordered pair is a solution to both original equations.\u201d It also says, \u201cSubstitute (4, \u22121) into 2x + y = 7 and x \u2013 2y = 6. Do they make both equations true? Yes!\u201d It then gives the equations. 2x + y = 7 becomes 2 times 4 + \u22121 = 7 which is 7 = 7. x \u2013 2y = 6 becomes 4 \u2013 2 times \u22121 = 6 which is 6 = 6. The row then says, \u201cThe solution is (4, \u22121).\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004g_img_new.jpg\" alt=\"The seventh row says, \u201cStep 7: Check that the ordered pair is a solution to both original equations.\u201d It also says, \u201cSubstitute (4, \u22121) into 2x + y = 7 and x \u2013 2y = 6. Do they make both equations true? Yes!\u201d It then gives the equations. 2x + y = 7 becomes 2 times 4 + \u22121 = 7 which is 7 = 7. x \u2013 2y = 6 becomes 4 \u2013 2 times \u22121 = 6 which is 6 = 6. The row then says, \u201cThe solution is (4, \u22121).\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-id1168345633560\" data-type=\"problem\">\n<p id=\"fs-id1168345197716\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}3x+y=5\\hfill \\\\ 2x-3y=7\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345427288\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345251140\">\\(\\left(2,-1\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\nThe steps are listed below for easy reference.\n<div id=\"fs-id1168345256923\" 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 solve a system of equations by elimination.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169752826769\" class=\"stepwise\" type=\"1\">\n \t<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n \t<li>Make the coefficients of one variable opposites.\n<ul id=\"fs-id1168345708283\" data-bullet-style=\"bullet\">\n \t<li>Decide which variable you will eliminate.<\/li>\n \t<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<\/ul>\n<\/li>\n \t<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n \t<li>Solve for the remaining variable.<\/li>\n \t<li>Substitute the solution from Step 4 into one of the original equations. Then solve for 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>\n<\/div>\n<p id=\"fs-id1168345677482\">First we\u2019ll do an example where we can eliminate one variable right away.<\/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-id1168345407940\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345429700\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}x+y=10\\hfill \\\\ x-y=12\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345230664\" 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-id1167824734317\" style=\"height: 305px; width: 100%;\" summary=\"This figure beings with two equations: x + y = 10 and x \u2013 y = 12. The figure then says, \u201cBoth equations are in standard form. The coefficients of y are already opposites. Add the two equations to eliminate y. The resulting equations has only 1 variable x. It then shows the equations added together. Thus x + y = 10 plus x \u2013 y = 12 becomes 2x = 22. The figure then instructs, \u201cSolve for x, the remaining variable.\u201d Thus x = 11. The figure then says, \u201cSolve for x = 11 into one of the original equations. Thus x + y = 10 becomes 11 + y = 10. The figure then says, \u201cSolve for the other variable, y.\u201d Thus y = -1. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (11, -1).\u201d It then states, \u201cCheck the ordered pair is a solution to both original equations. Thus x + y = 10 becomes 11 + (-1) = 10 and 10 = 10. x \u2013 y = 12 becomes 11 \u2013 (-1) = 12 and 12 = 12. The figure then states, \u201cThe solution is (11, -1).\" data-label=\"\">\n<tbody>\n<tr style=\"height: 19px;\">\n<td style=\"height: 19px; width: 50.8911%;\"><\/td>\n<td style=\"height: 19px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836530489\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">Both equations are in standard form.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">The coefficients of\u00a0<em data-effect=\"italics\">y<\/em>\u00a0are already opposites.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\" data-valign=\"top\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\">\n<\/span>The resulting equation has only 1 variable,\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836317852\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px; width: 50.8911%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>, the remaining variable.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 11 into one of the original equations.<\/td>\n<td style=\"height: 46px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167824590565\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\"><\/td>\n<td style=\"height: 14px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167829742244\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">Solve for the other variable,\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836787837\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\">The ordered pair is (11, \u22121).<\/td>\n<\/tr>\n<tr style=\"height: 94px;\">\n<td style=\"height: 94px; width: 50.8911%;\" data-valign=\"top\">Check that the ordered pair is a solution<span data-type=\"newline\">\n<\/span>to\u00a0<strong data-effect=\"bold\">both<\/strong>\u00a0original 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; 10\\hfill \\\\ \\hfill 11+\\left(-1\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 10\\hfill \\\\ \\hfill 10&amp; =\\hfill &amp; 10\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill x-y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill 11-\\left(-1\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 12\\hfill \\\\ \\hfill 12&amp; =\\hfill &amp; 12\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"height: 94px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\"><\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\">The solution is (11, \u22121).<\/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-id1168345291597\" data-type=\"problem\">\n<p id=\"fs-id1168345277703\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}2x+y=5\\hfill \\\\ x-y=4\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345255731\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345427021\">\\(\\left(3,-1\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\nIn the next example, we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.\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-id1168341961050\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345544062\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}3x-2y=-2\\hfill \\\\ 5x-6y=10\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345742599\" 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-id1167836360149\" style=\"width: 100%;\" summary=\"This figure begins with two equations: 3x -2y = -2 and 5x \u2013 6y = 10. The figure instructs, \u201cBoth equations are in standard form. None of the coefficients are opposites. We can make the coefficients of y opposites by multiplying the first equation by -3.\u201d The figure then shows the equations. The first is -3 times (3x \u2013 2y) = -3 times -2, and the second is 5x \u2013 6y = 10. The figure then instructs, \u201cSimplify.\u201d The two equations are -9x + 6y = 6 and 5x \u2013 6y = 10. The figure then says, \u201cAdd the two equations in eliminate y.\u201d The two equations added together becomes -4x = 16. The figure then says, \u201cSolve for the remaining variable x.\u201d Thus x = -4. The figure then instructs, \u201cSubstitute x = -4 into one of the original equations. Thus 3x \u2013 2y = -2 becomes 3 times -4 \u2013 2y = -2. The figure then instructs, \u201cSolve for y.\u201d The equation becomes -12 - 2y = 2 or -2y = 10. Thus y = -5. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (-4, -5).\u201d The figure then says, \u201cCheck that the ordered pair is a solution to both original equations.\u201d Thus 3x -2y = -2 becomes 3 times -4 - 2 times -5 = -2 or -12 +10 = -2 or -2y = -2. It also shows that 5x \u2013 6y = 10 becomes 3 times -4 \u2013 6 times -5 = 10 or -20 + 30 = 10. Thus 10 = 10. The figure then says, \u2018The solutions is (-4, -5).\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 508.047px;\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836609520\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Both equations are in standard form.<\/td>\n<td style=\"width: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">None of the coefficients are opposites.<\/td>\n<td style=\"width: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">We can make the coefficients of\u00a0<em data-effect=\"italics\">y<\/em>\u00a0opposites by multiplying<span data-type=\"newline\">\n<\/span>the first equation by \u22123.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167833316764\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167833057104\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836596232\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Solve for the remaining variable,\u00a0<em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\">\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= \u22124 into one of the original equations.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836662540\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167829597819\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167829619693\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167833274679\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1167836536250\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\">The ordered pair is (\u22124, \u22125).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Check that the ordered pair is a solution to<span data-type=\"newline\">\n<\/span>both original equations.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill 3x-2y&amp; =\\hfill &amp; -2\\hfill \\\\ \\hfill 3\\left(-4\\right)-2\\left(-5\\right)&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill \\\\ \\hfill -12+10&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill \\\\ \\hfill -2y&amp; =\\hfill &amp; -2\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill 5x-6y&amp; =\\hfill &amp; 10\\hfill \\\\ \\hfill 3\\left(-4\\right)-6\\left(-5\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 10\\hfill \\\\ \\hfill -20+30&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: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\">The solution is (\u22124, \u22125).<\/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 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n\nSolve the system by elimination. \\(\\left\\{\\begin{array}{c}4x-3y=1\\hfill \\\\ 5x-9y=-4\\hfill \\end{array}\\)\n\n<\/div>\n<div id=\"fs-id1168345212163\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345734996\">\\(\\left(1,1\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345507881\">Now we\u2019ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.<\/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-id1168345556370\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345744930\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}4x-3y=9\\hfill \\\\ 7x+2y=-6\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345284539\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\nIn this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by a constant to get the opposites.<span data-type=\"newline\">\n<\/span>\n<table id=\"fs-id1167836530249\" style=\"width: 100%;\" summary=\"This figure begins with two equations: 4x \u2013 3y = 9 and 7x + 2y = -6. The figure then says, \u201cBoth equations are in standard form. To get opposite coefficients of y, we will multiply the first equation by 2 and the second equation by 3.\u201d It then shows the equations as 2 times (4x \u2013 3y) = 2 times 9 and 3 times (7x + 2y) = 3 times -6. The figure then says, \u201cSimplify.\u201d The equations then become 8x \u2013 6y = 18 and 21x + 6y = -18. The figure then says, \u201cAdd the two equations to eliminate y. After adding, the answer is 39x = 0. The figure then says, \u201cSolve for x.\u201d Thus, x = 0. The figure then reads, \u201cSubstitute x = 0 into one of the original equations.\u201d Thus 7x +2y = -6 becomes 7 times 0 + 2y = -6. The figure then says, \u201cSolve for y.\u201d It then says, 2y = -6 and thus 2y = -3. The figure then reads, \u201cWrite the solution as an ordered pair. The ordered pair is (0, -3).\u201d The figure then instructs, \u201cCheck that the ordered pair is a solution to both original equations. Thus 4x \u2013 3y = 9 becomes 4 times 0 \u2013 3 times -3 = 9 or 9 = 9. Thus 7x + 2y = -6 becomes 7 times 0 + 2 times -3 = -6 or -6 = -6. The figure then says, \u201cThe solution is (0, -3).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836516310\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Both equations are in standard form. To get opposite<span data-type=\"newline\">\n<\/span>coefficients of\u00a0<em data-effect=\"italics\">y<\/em>, we will multiply the first equation by 2<span data-type=\"newline\">\n<\/span>and the second equation by 3.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167829693355\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167829850392\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"middle\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167832930182\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 0 into one of the original equations.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836349138\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836688612\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836306660\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836503981\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\">The ordered pair is (0, \u22123).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Check that the ordered pair is a solution to<span data-type=\"newline\">\n<\/span><strong data-effect=\"bold\">both<\/strong>\u00a0original equations.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill 4x-3y&amp; =\\hfill &amp; 9\\hfill \\\\ \\hfill 4\\left(0\\right)-3\\left(-3\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 9\\hfill \\\\ \\hfill 9&amp; =\\hfill &amp; 9\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill 7x+2y&amp; =\\hfill &amp; -6\\hfill \\\\ \\hfill 7\\left(0\\right)+2\\left(-3\\right)&amp; \\stackrel{?}{=}\\hfill &amp; -6\\hfill \\\\ \\hfill -6&amp; =\\hfill &amp; -6\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 52.4752%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\">The solution is (0, \u22123).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345367998\"><\/p>\n\n<\/div>\n<\/div>\n<\/div>\nWhat other constants could we have chosen to eliminate one of the variables? Would the solution be the same?\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-id1168345406843\" data-type=\"problem\">\n<p id=\"fs-id1168345360570\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}3x-4y=-9\\hfill \\\\ 5x+3y=14\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345414924\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345511147\">\\(\\left(1,3\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345419206\">When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by its LCD.<\/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-id1168341960861\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345292398\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}x+\\frac{1}{2}y=6\\hfill \\\\ \\frac{3}{2}x+\\frac{2}{3}y=\\frac{17}{2}\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345404754\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1168345217762\">In this example, both equations have fractions. Our first step will be to multiply each equation by its LCD to clear the fractions.<span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167836409014\" class=\"unnumbered unstyled can-break\" summary=\"This figure beings with two equations: x + (1\/2)y = 6 and (3\/2)x + (2\/3)y = 17\/2. The figure says, \u201cTo clear the fractions multiply each equation by its LCD.\u201d It then shows the equations as 2 times (x + (1\/2)y) = 2 times 6 and 6 times ((3\/2)x + (2\/3)y) = 6 times (17\/2). The figure then says, \u201cSimplify.\u201d The equations then become 2x + y = 12 and 9x + 4y = 51. The figure then says, \u201cNow we are ready to eliminate one of the variables. Notice that both equations are in standard form. We can eliminate y multiplying the top equation by -4.\u201d It then shows -4 times (2x + y) = -4 times 12 and 9x + 4y = 51. The figure then says, \u201cSimplify and add.\u201d The equations added are thus -8x \u2013 4y = -48 plus 9x + 4y = 51 which gives x = 3. The figure then says, \u201cSubstitute x = 3 into one of the original equations. Solve for y.\u201d Thus x + (1\/2)y = 6 becomes 3 + (1\/2)y = 6. This becomes (1\/2)y = 3 or y = 6. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (3, 6). The figure then says, \u201cCheck the ordered pair is a solution to both original equations. Thus x + (1\/2)y = 6 becomes 3 + (1\/2) times 6 = 6 or 3 + 6 = 6. Thus 6 = 6. The second equation is (3\/2)x + (2\/3)y = 17\/2 or (3\/2) times 3 + (2\/3) times 6 = 17\/2. This becomes 9\/2 + 4 = 17\/2 or 9\/2 + 8\/2 = 17\/2. Thus 17\/2 = 17\/2. The figure then says, \u201cThe solution is (3, 6).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836688835\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">To clear the fractions, multiply each equation by its LCD.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836502141\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836352044\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Now we are ready to eliminate one of the variables. Notice that<span data-type=\"newline\">\n<\/span>both equations are in standard form.<\/td>\n<td style=\"width: 289px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">We can eliminate\u00a0<em data-effect=\"italics\">y<\/em>\u00a0multiplying the top equation by \u22124.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836560117\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Simplify and add.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 into one of the original equations.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836300298\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836342400\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836388312\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836627560\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\">The ordered pair is (3, 6).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Check that the ordered pair is a solution<span data-type=\"newline\">\n<\/span>to\u00a0<strong data-effect=\"bold\">both<\/strong>\u00a0original equations.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>\\(\\begin{array}{cccc}\\begin{array}{ccc}\\hfill x+\\frac{1}{2}y&amp; =\\hfill &amp; 6\\hfill \\\\ \\hfill 3+\\frac{1}{2}\\left(6\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 6\\hfill \\\\ \\hfill 3+6&amp; \\stackrel{?}{=}\\hfill &amp; 6\\hfill \\\\ \\hfill 6&amp; =\\hfill &amp; 6\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}\\hfill \\frac{3}{2}x+\\frac{2}{3}y&amp; =\\hfill &amp; \\frac{17}{2}\\hfill \\\\ \\hfill \\frac{3}{2}\\left(3\\right)+\\frac{2}{3}\\left(6\\right)&amp; \\stackrel{?}{=}\\hfill &amp; \\frac{17}{2}\\hfill \\\\ \\hfill \\frac{9}{2}+4&amp; \\stackrel{?}{=}\\hfill &amp; \\frac{17}{2}\\hfill \\\\ \\hfill \\frac{9}{2}+\\frac{8}{2}&amp; \\stackrel{?}{=}\\hfill &amp; \\frac{17}{2}\\hfill \\\\ \\hfill \\frac{17}{2}&amp; =\\hfill &amp; \\frac{17}{2}\\phantom{\\rule{0.2em}{0ex}}\u2713\\hfill \\end{array}\\end{array}\\)<\/td>\n<td style=\"width: 289px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\">The solution is (3, 6).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345428888\" data-type=\"problem\">\n\nSolve the system by elimination. \\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-\\frac{1}{2}y=1\\hfill \\\\ \\frac{3}{4}x-y=\\frac{5}{2}\\hfill \\end{array}\\)\n\n<\/div>\n<div id=\"fs-id1168345286997\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345290805\">\\(\\left(6,2\\right)\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345357619\">When we were solving systems of linear equations by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.<\/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-id1168345287937\" data-type=\"note\">\n<div id=\"fs-id1168345213647\" data-type=\"exercise\">\n<div id=\"fs-id1168345645051\" data-type=\"problem\">\n<p id=\"fs-id1168345550525\">Solve the system by elimination:<\/p>\na) \\(\\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ y=3-\\frac{3}{4}x\\hfill \\end{array}\\)\n\nb) \\(\\left\\{\\begin{array}{c}5x-3y=15\\hfill \\\\ y=-5+\\frac{5}{3}x\\hfill \\end{array}\\)\n\nc) <span style=\"text-align: initial; font-size: 0.9em; word-spacing: normal;\">\\(\\left\\{\\begin{array}{c}x+2y=6\\hfill \\\\ y=-\\frac{1}{2}x+3\\hfill \\end{array}\\)<\/span>\n\nd)\\(\\left\\{\\begin{array}{c}-6x+15y=10\\hfill \\\\ 2x-5y=-5\\hfill \\end{array}\\)\n\n&nbsp;\n\n<\/div>\n<div id=\"fs-id1168345346541\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-149\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 410.094px;\">a)<\/td>\n<td style=\"width: 440.125px;\">\\(\\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ y=3-\\frac{3}{4}x\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Write the second equation in standard form.<\/td>\n<td style=\"width: 440.125px;\">\\(\\left\\{\\begin{array}{ccc}\\hfill 3x+4y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill \\frac{3}{4}x+y&amp; =\\hfill &amp; 3\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Clear the fractions by multiplying the second equation by 4.<\/td>\n<td style=\"width: 440.125px;\">\\(\\left\\{\\begin{array}{ccc}\\hfill 3x+4y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill 4\\left(\\frac{3}{4}x+y\\right)&amp; =\\hfill &amp; 4\\left(3\\right)\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Simplify.<\/td>\n<td style=\"width: 440.125px;\">\\(\\left\\{\\begin{array}{ccc}\\hfill 3x+4y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill 3x+4y&amp; =\\hfill &amp; 12\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">To eliminate a variable, we multiply the second equation by \\(-1\\).\u00a0<span data-type=\"newline\">\n<\/span>Simplify and add.<\/td>\n<td style=\"width: 440.125px;\">\\(\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}{\\left\\{\\begin{array}{ccc}\\hfill 3x+4y&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill -3x-4y&amp; =\\hfill &amp; -12\\hfill \\end{array}}\\hfill \\\\ \\hfill 0=0\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">This is a true statement. The equations are consistent but dependent. Their graphs would be the same line. The system has infinitely many solutions.<\/td>\n<td style=\"width: 440.125px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">After we cleared the fractions in the second equation, did you notice that the two equations were the same? That means we have coincident lines.<\/td>\n<td style=\"width: 440.125px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">b)<\/td>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1168345213649\" data-type=\"problem\">\n<p id=\"fs-id1168345539011\">\\(\\left\\{\\begin{array}{c}5x-3y=15\\hfill \\\\ y=-5+\\frac{5}{3}x\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168341917062\" data-type=\"solution\">\n<p id=\"fs-id1168345415533\"><\/p>\n\n<\/div><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">infinitely many solutions<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1168345213649\" data-type=\"problem\"><\/div>\n<div id=\"fs-id1168341917062\" data-type=\"solution\">\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">c)<\/td>\n<td style=\"width: 50%;\">\\(\\left\\{\\begin{array}{c}x+2y=6\\hfill \\\\ y=-\\frac{1}{2}x+3\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">infinitely many solutions<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341961563\" data-type=\"note\">\n<div data-type=\"exercise\">\n<div id=\"fs-id1168345511489\" data-type=\"problem\"><\/div>\n<div id=\"fs-id1168345450687\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<table id=\"eip-353\" class=\"unnumbered unstyled\" style=\"height: 192px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 429px;\">d)<\/td>\n<td style=\"height: 16px; width: 420px;\">\\(\\left\\{\\begin{array}{c}-6x+15y=10\\hfill \\\\ 2x-5y=-5\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 429px;\">The equations are in standard form.<\/td>\n<td style=\"height: 32px; width: 420px;\">\\(\\left\\{\\begin{array}{ccc}\\hfill -6x+15y&amp; =\\hfill &amp; 10\\hfill \\\\ \\hfill 2x-5y&amp; =\\hfill &amp; -5\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 429px;\">Multiply the second equation by 3 to eliminate a variable.<\/td>\n<td style=\"height: 32px; width: 420px;\">\\(\\left\\{\\begin{array}{ccc}\\hfill -6x+15y&amp; =\\hfill &amp; 10\\hfill \\\\ \\hfill 3\\left(2x-5y\\right)&amp; =\\hfill &amp; 3\\left(-5\\right)\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 429px;\">Simplify and add.<\/td>\n<td style=\"height: 48px; width: 420px;\">\\(\\begin{array}{c}\\text{}{\\left\\{\\begin{array}{ccc}\\hfill -6x+15y&amp; =\\hfill &amp; \\phantom{\\rule{0.5em}{0ex}}10\\hfill \\\\ \\hfill 6x-15y&amp; =\\hfill &amp; -15\\hfill \\end{array}}\\\\ \\hfill 0\\ne -5\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 429px;\">This statement is false. The equations are inconsistent and so their graphs would be parallel lines.<\/td>\n<td style=\"height: 48px; width: 420px;\"><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 429px;\">The system does not have a solution.<\/td>\n<td style=\"height: 16px; width: 420px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341952290\" data-type=\"problem\">\n<p id=\"fs-id1168341952292\">Solve the system by elimination. \\(\\left\\{\\begin{array}{c}-3x+2y=8\\hfill \\\\ 9x-6y=13\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345447658\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345447660\">no solution<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916882\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Elimination<\/h1>\n<p id=\"fs-id1168341916887\">Some applications problems translate directly into equations in standard form, so we will use the elimination method to solve them. As before, we use our Problem Solving Strategy to help us stay focused and organized.<\/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 id=\"fs-id1168345255373\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n\nThe sum of two numbers is 39. Their difference is 9. Find the numbers.\n\n<\/div>\n<div 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=\"eip-269\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td style=\"width: 524.562px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 524.562px;\">We are looking for two numbers.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 3. Name<\/strong>\u00a0what we are looking for.\u00a0<span data-type=\"newline\">\n<\/span>Choose a variable to represent that quantity.<\/td>\n<td style=\"width: 524.562px;\">Let \\(n=\\) the first number.\u00a0<span data-type=\"newline\">\n<\/span>\\(m=\\) the second number.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 4. Translate<\/strong>\u00a0into a system of equations.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>The system is:<\/td>\n<td style=\"width: 524.562px;\">The sum of two numbers is 39.\u00a0<span data-type=\"newline\">\n<\/span>\\(n+m=39\\)<span data-type=\"newline\">\n<\/span>Their difference is 9.\u00a0<span data-type=\"newline\">\n<\/span>\\(\\begin{array}{c}\\hfill n-m=9\\hfill \\\\ \\hfill \\left\\{\\begin{array}{c}n+m=39\\hfill \\\\ n-m=9\\hfill \\end{array}\\hfill \\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 5. Solve<\/strong>\u00a0the system of equations.\u00a0<span data-type=\"newline\">\n<\/span>To solve the system of equations, use elimination.\u00a0<span data-type=\"newline\">\n<\/span>The equations are in standard form and the coefficients of \\(m\\) are opposites. Add.\u00a0<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Solve for \\(n\\).\u00a0<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Substitute \\(n=24\\) into one of the original equations and solve for \\(m\\).<\/td>\n<td style=\"width: 524.562px;\">\\(\\begin{array}{c}\\hfill \\text{}{\\left\\{\\begin{array}{c}n+m=39\\hfill \\\\ n-m=9\\hfill \\end{array}}\\hfill \\\\ \\hfill 2n\\phantom{\\rule{1.8em}{0ex}}=48\\hfill \\\\ \\\\ \\hfill \\phantom{\\rule{2.21em}{0ex}}n=24\\hfill \\\\ \\hfill n+m=39\\\\ \\hfill 24+m=39\\\\ \\hfill m=15\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td style=\"width: 524.562px;\">Since \\(24+15=39\\) and \\(24-15=9\\), the answers check.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td style=\"width: 524.562px;\">The numbers are 24 and 15.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345577233\" data-type=\"problem\">\n<p id=\"fs-id1168345577235\">The sum of two numbers is 42. Their difference is 8. Find the numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1168345577239\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345577241\">The numbers are 25 and 17.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\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-id1168345542428\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345542430\">Joe stops at a burger restaurant every day on his way to work. Monday he had one order of medium fries and two small sodas, which had a total of 620 calories. Tuesday he had two orders of medium fries and one small soda, for a total of 820 calories. How many calories are there in one order of medium fries? How many calories in one small soda?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345665125\" 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-id1167836534642\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what we are looking for. We are looking for the number of calories in one order of medium fires and in one small soda. Step 3. Name what we are looking for. Let f = the number of calories in 1 order of medium fries. s = the number of calories in 1 small soda. Step 4. Translate into a system of equations: one medium fries and two small sodas had a total of 620 calories. f + 2s = 620. Two medium fries and one small soda had a total of 820 calories. 2f + s = 820. Our syste is f + 2s = 620 and 2f + s = 820. Step 5. Solve the system of equations. To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of f, multiply the top equation by -2.\u201d The equations are -2(f + 2s) = -2 times 620 and 2f + s =820. The figure then says, \u201cSimplify and add.\u201d Thus -2f \u2013 4s = -1240 plus 2f + s = 820 equals -3s = -420. The figure then says, \u201cSolve for s.\u201d Thus s = 140. The figure then reads, \u201cSubstitute s = 140 into one of the original equations and then solve for f. Thus, f + 2s = 620 becomes f + 2 times 140 = 620 or f +280 = 620. Thus f = 340. The figure then reads, \u201cStep 6. Check the answer. Verify that these numbers make sense in the problem and that they are solutions to both equations. We leave this to you! Step 7. Answer the question. The small soda has 140 calories and the fries have 340 calories.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td style=\"width: 57.2727%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">We are looking for the number of<span data-type=\"newline\">\n<\/span>calories in one order of medium fries<span data-type=\"newline\">\n<\/span>and in one small soda.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">Let\u00a0<em data-effect=\"italics\">f<\/em>\u00a0= the number of calories in<span data-type=\"newline\">\n<\/span>1 order of medium fries.<span data-type=\"newline\">\n<\/span>\u2003\u00a0\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= the number of calories in<span data-type=\"newline\">\n<\/span>1 small soda.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations:<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">one medium fries and two small sodas had a<span data-type=\"newline\">\n<\/span>total of 620 calories<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836492134\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">two medium fries and one small soda had a<span data-type=\"newline\">\n<\/span>total of 820 calories.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167824658652\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Our system is:<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836319166\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\">\n<\/span>To solve the system of equations, use<span data-type=\"newline\">\n<\/span>elimination. The equations are in standard<span data-type=\"newline\">\n<\/span>form. To get opposite coefficients of\u00a0<em data-effect=\"italics\">f<\/em>,<span data-type=\"newline\">\n<\/span>multiply the top equation by \u22122.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836406845\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Simplify and add.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167833356376\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">s<\/em>.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167833338872\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Substitute\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 140 into one of the original<span data-type=\"newline\">\n<\/span>equations and then solve for\u00a0<em data-effect=\"italics\">f<\/em>.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836318741\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836533791\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836507422\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836549057\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">Verify that these numbers make sense<span data-type=\"newline\">\n<\/span>in the problem and that they are<span data-type=\"newline\">\n<\/span>solutions to both equations.<span data-type=\"newline\">\n<\/span>We leave this to you!<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">The small soda has 140 calories and<span data-type=\"newline\">\n<\/span>the fries have 340 calories.<\/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 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341862434\" data-type=\"problem\">\n<p id=\"fs-id1168341862436\">Malik stops at the grocery store to buy a bag of diapers and 2 cans of formula. He spends a total of \\$37. The next week he stops and buys 2 bags of diapers and 5 cans of formula for a total of \\$87. How much does a bag of diapers cost? How much is one can of formula?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345229866\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345229868\">The bag of diapers costs ?11 and the can of formula costs ?13.<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345416871\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/h1>\n<p id=\"fs-id1168345550343\">When you will have to solve a system of linear equations in a later math class, you will usually not be told which method to use. You will need to make that decision yourself. So you\u2019ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.<\/p>\n<span id=\"fs-id1169752961745\" data-type=\"media\" data-alt=\"This table has two rows and three columns. The first row labels the columns as \u201cGraphing,\u201d \u201cSubstitution,\u201d and \u201cElimination.\u201d Under \u201cGraphing\u201d it says, \u201cUse when you need a picture of the situation.\u201d Under \u201cSubstitution\u201d it says, \u201cUse when one equation is already solved for one variable.\u201d Under \u201cElimination\u201d it says, \u201cUse when the equations are in standard form.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_021_img.jpg\" alt=\"This table has two rows and three columns. The first row labels the columns as \u201cGraphing,\u201d \u201cSubstitution,\u201d and \u201cElimination.\u201d Under \u201cGraphing\u201d it says, \u201cUse when you need a picture of the situation.\u201d Under \u201cSubstitution\u201d it says, \u201cUse when one equation is already solved for one variable.\u201d Under \u201cElimination\u201d it says, \u201cUse when the equations are in standard form.\u201d\" data-media-type=\"image\/jpeg\"><\/span>\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-id1168341863749\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168341863751\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1168341863756\"><span class=\"token\">a) <\/span>\\(\\left\\{\\begin{array}{c}3x+8y=40\\hfill \\\\ 7x-4y=-32\\hfill \\end{array}\\)<\/p>\nb) \\(\\left\\{\\begin{array}{c}5x+6y=12\\hfill \\\\ y=\\frac{2}{3}x-1\\hfill \\end{array}\\)\n\n<strong style=\"font-size: 16.8px; word-spacing: normal;\">Solution<\/strong>\n\n<\/div>\n<div id=\"fs-id1168341923301\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1163872390948\"><span class=\"token\">a) <\/span>\\(\\begin{array}{ccc}&amp; &amp; \\left\\{\\begin{array}{c}3x+8y=40\\hfill \\\\ 7x-4y=-32\\hfill \\end{array}\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span>Since both equations are in standard form, using elimination will be most convenient.<span data-type=\"newline\">\n<\/span><\/p>\n<span class=\"token\">b)<\/span>\u00a0\\(\\begin{array}{ccc}&amp; &amp; \\left\\{\\begin{array}{c}5x+6y=12\\hfill \\\\ y=\\frac{2}{3}x-1\\hfill \\end{array}\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span>Since one equation is already solved for\u00a0<em data-effect=\"italics\">y<\/em>, using substitution will be most convenient.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345580362\" data-type=\"problem\">\n<p id=\"fs-id1168345580364\">For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1168345580369\">a) \\(\\left\\{\\begin{array}{c}4x-5y=-32\\hfill \\\\ 3x+2y=-1\\hfill \\end{array}\\)<\/p>\nb) \\(\\left\\{\\begin{array}{c}x=2y-1\\hfill \\\\ 3x-5y=-7\\hfill \\end{array}\\)\n\n<\/div>\n<div id=\"fs-id1168345443554\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345443556\">a) Since both equations are in standard form, using elimination will be most convenient.<\/p>\nb) Since one equation is already solved for \\(x\\), using substitution will be most convenient.\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 linear equations by elimination.<\/span>\n<div class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1168345665200\" data-display=\"block\">\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination1\">Instructional Video-Solving Systems of Equations by Elimination<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination2\">Instructional Video-Solving by Elimination<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination3\">Instructional Video-Solving Systems by Elimination<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168345723908\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">To Solve a System of Equations by Elimination<\/strong>\n<ol id=\"fs-id1169750655024\" class=\"stepwise\" type=\"1\">\n \t<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n \t<li>Make the coefficients of one variable opposites.\n<ul id=\"fs-id1171790125566\" data-bullet-style=\"circled\">\n \t<li>Decide which variable you will eliminate.<\/li>\n \t<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<\/ul>\n<\/li>\n \t<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n \t<li>Solve for the remaining variable.<\/li>\n \t<li>Substitute the solution from Step 4 into one of the original equations. Then solve for 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<\/li>\n<\/ul>\n<div id=\"fs-id1168341962936\" class=\"practice-perfect\" data-depth=\"2\">\n<h1 data-type=\"title\">4.3 Exercise Set<\/h1>\n<p id=\"fs-id1169747646746\">In the following exercises, solve the systems of equations by elimination.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\\(\\left\\{\\begin{array}{c}-3x+y=-9\\hfill \\\\ x-2y=-12\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x-y=-7\\hfill \\\\ 4x+2y=-6\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+y=-8\\hfill \\\\ x-y=-6\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-7x+6y=-10\\hfill \\\\ x-6y=22\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}5x+2y=1\\hfill \\\\ -5x-4y=-7\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x-4y=-11\\hfill \\\\ x-2y=-5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}6x-5y=-75\\hfill \\\\ -x-2y=-13\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x-5y=7\\hfill \\\\ 3x-y=17\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}7x+y=-4\\hfill \\\\ 13x+3y=4\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x-5y=-9\\hfill \\\\ 5x+2y=16\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}4x+7y=14\\hfill \\\\ -2x+3y=32\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x+8y=-3\\hfill \\\\ 2x+5y=-3\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x+8y=67\\hfill \\\\ 5x+3y=60\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-y=-3\\hfill \\\\ x+\\frac{5}{2}y=2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x+\\frac{1}{3}y=-1\\hfill \\\\ \\frac{1}{2}x-\\frac{1}{3}y=-2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+y=3\\hfill \\\\ 6x+3y=9\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-3x-y=8\\hfill \\\\ 6x+2y=-16\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x+2y=6\\hfill \\\\ -6x-4y=-12\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}-11x+12y=60\\hfill \\\\ -22x+24y=90\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}5x-3y=15\\hfill \\\\ y=\\frac{5}{3}x-2\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<p id=\"fs-id1169747673556\">In the following exercises, translate to a system of equations and solve.<\/p>\n\n<ol start=\"21\">\n \t<li>The sum of two numbers is 65. Their difference is 25. Find the numbers<span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\">.<\/span><\/li>\n \t<li>The sum of two numbers is \u221227. Their difference is \u221259. Find the numbers.<\/li>\n \t<li>Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for \\$114 or she is able to buy 2 shirts and 4 sweaters for \\$164. How much does a shirt cost? How much does a sweater cost?<\/li>\n \t<li>The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?<\/li>\n<\/ol>\n<p id=\"fs-id1169752961726\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p>\n\n<ol start=\"25\">\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\left\\{\\begin{array}{c}8x-15y=-32\\hfill \\\\ 6x+3y=-5\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x=4y-3\\hfill \\\\ 4x-2y=-6\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\left\\{\\begin{array}{c}y=4x+9\\hfill \\\\ 5x-2y=-21\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}9x-4y=24\\hfill \\\\ 3x+5y=-14\\hfill \\end{array}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>Norris can row 3 miles upstream against the current in the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. \\(\\left\\{\\begin{array}{c}r-c=3\\hfill \\\\ r+c=5\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>for \\(r\\), his rowing speed in still water.<\/li>\n \t<li>Then solve for \\(c\\), the speed of the river current.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1168345487655\" data-type=\"exercise\">\n<div id=\"fs-id1168345487657\" data-type=\"problem\">\n<p id=\"fs-id1168345487659\"><span data-type=\"newline\"><span style=\"background-color: initial; font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Answers:<\/span>\n<\/span><\/p>\n\n<ol class=\"twocolumn\">\n \t<li>(6, 9)<\/li>\n \t<li>\\(\\left(-2,1\\right)\\)<\/li>\n \t<li>\\(\\left(-7,-1\\right)\\)<\/li>\n \t<li>\\(\\left(-2,-4\\right)\\)<\/li>\n \t<li>\\(\\left(-1,3\\right)\\)<\/li>\n \t<li>\\(\\left(-1,2\\right)\\)<\/li>\n \t<li>\\(\\left(-5,9\\right)\\)<\/li>\n \t<li>(6, 1)<\/li>\n \t<li>\\(\\left(-2,10\\right)\\)<\/li>\n \t<li>(2, 3)<\/li>\n \t<li>\\(\\left(-7,6\\right)\\)<\/li>\n \t<li>\\(\\left(-9,3\\right)\\)<\/li>\n \t<li>(9, 5)<\/li>\n \t<li>\\(\\left(-3,2\\right)\\)<\/li>\n \t<li>\\(\\left(-2,3\\right)\\)<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>infinitely many solutions<\/li>\n \t<li>inconsistent, no solution<\/li>\n \t<li>inconsistent, no solution<\/li>\n \t<li>The numbers are 20 and 45.<\/li>\n \t<li>The numbers are 16 and \u221243.<\/li>\n \t<li>A shirt costs \\$16 and a sweater costs \\$33.<\/li>\n \t<li>There are 860 mg in a hot dog. There are 1,000 mg in a cup of cottage cheese.<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>elimination<\/li>\n \t<li>substitution<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>substitution<\/li>\n \t<li>elimination<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(r=4\\)<\/li>\n \t<li>\\(c=1\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\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 elimination<\/li>\n<li>Solve applications of systems of equations by elimination<\/li>\n<li>Choose the most convenient method to solve a system of linear equations<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345297121\" class=\"be-prepared\" data-type=\"note\"><\/div>\n<p id=\"fs-id1168345215927\">We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.<\/p>\n<p id=\"fs-id1168345270702\">The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we\u2019ll do with the elimination method, too, but we\u2019ll have a different way to get there.<\/p>\n<div id=\"fs-id1168345742458\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Equations by Elimination<\/h1>\n<p id=\"fs-id1168341917939\">The <span class=\"no-emphasis\" data-type=\"term\">Elimination Method<\/span> is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.<\/p>\n<p id=\"fs-id1168345415661\">For any expressions <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em>,<\/p>\n<div id=\"fs-id1168345260014\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b546f4b1aed6f340a4e86395750d423_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;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#99;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#43;&#99;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#98;&#43;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"175\" style=\"vertical-align: -24px;\" \/><\/div>\n<p id=\"fs-id1168345342752\">To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.<\/p>\n<p id=\"fs-id1168345237002\">Notice how that works when we add these two equations together:<\/p>\n<div id=\"fs-id1168345213321\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d83333c34c98940ed8e8680ed2b5f506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#50;&#120;&#45;&#121;&#61;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"83\" style=\"vertical-align: -22px;\" \/><\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><span style=\"font-size: 14pt; text-align: initial;\">The <\/span><em style=\"font-size: 14pt; text-align: initial;\" data-effect=\"italics\">y<\/em><span style=\"font-size: 14pt; text-align: initial;\">\u2019s add to zero and we have one equation with one variable.<\/span><\/div>\n<\/div>\n<div id=\"fs-id1168345742458\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1168345261092\">Let\u2019s try another one:<\/p>\n<div id=\"fs-id1168345418800\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f1553a05809aacb29670b75135dff088_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;&#52;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168345376612\">This time we don\u2019t see a variable that can be immediately eliminated if we add the equations.<\/p>\n<p id=\"fs-id1168345558535\">But if we multiply the first equation by \u22122, we will make the coefficients of <em data-effect=\"italics\">x<\/em> opposites. We must multiply every term on both sides of the equation by \u22122.<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows two equations. The first is negative 2 times x plus 4y in parentheses equals negative 2 times 2. The second is 2x + 5y = negative 2. This figure shows two equations. The first is negative 2x minus 8y = negative 4. The second is 2x + 5y = -negative 2.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_03_002_img_new.jpg\" alt=\"This figure shows two equations. The first is negative 2 times x plus 4y in parentheses equals negative 2 times 2. The second is 2x + 5y = negative 2. This figure shows two equations. The first is negative 2x minus 8y = negative 4. The second is 2x + 5y = -negative 2.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168341959314\">Now we see that the coefficients of the <em data-effect=\"italics\">x<\/em> terms are opposites, so <em data-effect=\"italics\">x<\/em> will be eliminated when we add these two equations.<\/p>\n<p id=\"fs-id1168345692963\">Add the equations yourself\u2014the result should be \u22123<em data-effect=\"italics\">y<\/em> = \u22126. And that looks easy to solve, doesn\u2019t it? Here is what it would look like.<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows two equations being added together. The first is negative 2x \u2013 8y = \u22124 and 2x plus 5y = negative 2. The answer is negative 3y = negative 6.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_010_img_new.jpg\" alt=\"This figure shows two equations being added together. The first is negative 2x \u2013 8y = \u22124 and 2x plus 5y = negative 2. The answer is negative 3y = negative 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345298294\">We\u2019ll do one more:<\/p>\n<div id=\"fs-id1168345262801\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b9863ad85bc967d4307c2bbeeeb9ace_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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#53;&#121;&#61;&#45;&#55;&#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;\" \/><\/div>\n<p id=\"fs-id1168345192582\">It doesn\u2019t appear that we can get the coefficients of one variable to be opposites by multiplying one of the equations by a constant, unless we use fractions. So instead, we\u2019ll have to multiply both equations by a constant.<\/p>\n<p>We can make the coefficients of <em data-effect=\"italics\">x<\/em> be opposites if we multiply the first equation by 3 and the second by \u22124, so we get 12<em data-effect=\"italics\">x<\/em> and \u221212<em data-effect=\"italics\">x<\/em>.<\/p>\n<p><span id=\"fs-id1168345290693\" data-type=\"media\" data-alt=\"This figure shows two equations. The first is 3 times 4x minus 3y in parentheses equals 3 times 10. The second is negative 4 times 3x plus 5y in parentheses equals negative 4 times negative 7.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_003_img_new.jpg\" alt=\"This figure shows two equations. The first is 3 times 4x minus 3y in parentheses equals 3 times 10. The second is negative 4 times 3x plus 5y in parentheses equals negative 4 times negative 7.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345458655\">This gives us these two new equations:<\/p>\n<div id=\"fs-id1168341892603\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ef739fabcdb3a9980ef80b256388d762_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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#120;&#45;&#57;&#121;&#61;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#49;&#50;&#120;&#45;&#50;&#48;&#121;&#61;&#50;&#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=\"151\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168345461401\">When we add these equations,<\/p>\n<div id=\"fs-id1168345459263\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b58272c1532845dcf1b553792eea62e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#120;&#45;&#57;&#121;&#61;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#45;&#49;&#50;&#120;&#45;&#50;&#48;&#121;&#61;&#50;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#57;&#121;&#61;&#53;&#56;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"162\" style=\"vertical-align: -26px;\" \/><\/div>\n<p id=\"fs-id1169747561781\">the <em data-effect=\"italics\">x<\/em>\u2019s are eliminated and we just have \u221229<em data-effect=\"italics\">y<\/em> = 58.<\/p>\n<p>Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.<\/p>\n<p>Now we\u2019ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.<\/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 style=\"font-size: 16.8px;\" data-type=\"title\">How to Solve a System of Equations by Elimination<\/div>\n<div id=\"fs-id1168345228558\" style=\"font-size: 16.8px;\" data-type=\"exercise\">\n<div id=\"fs-id1168345427214\" data-type=\"problem\">\n<p id=\"fs-id1168345376536\">Solve the system by elimination. <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<\/div>\n<div id=\"fs-id1168341852589\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1168341955857\" data-type=\"media\" data-alt=\"This figure has seven rows and three columns. The first row reads, \u201cStep 1. Write both equations in standard form. If any coefficients are fractions, clear them.\u201d It also says, \u201cBoth equations are in standard form, A x + B y = C. There are no fractions.\u201d It also gives the two equations as 2x + y = 7 and x \u2013 2y = 6.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004a_img_new.jpg\" alt=\"This figure has seven rows and three columns. The first row reads, \u201cStep 1. Write both equations in standard form. If any coefficients are fractions, clear them.\u201d It also says, \u201cBoth equations are in standard form, A x + B y = C. There are no fractions.\u201d It also gives the two equations as 2x + y = 7 and x \u2013 2y = 6.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345199327\" data-type=\"media\" data-alt=\"The second row reads, \u201cStep 2: Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.\u201d It also says, \u201cWe can eliminate the y\u2019s by multiplying the first equation by 2. Multiply both sides of 2x + y = 7 by 2.\u201d It also shows the steps with equations. Initially the equations are ex + y = 7 and x \u2013 2y = 6. Then they become 2(2x + y) = 2 times 7 and x \u2013 2y = 6. They then become 4x + 2y = 14 and x \u2013 2y = 6.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004b_img_new.jpg\" alt=\"The second row reads, \u201cStep 2: Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.\u201d It also says, \u201cWe can eliminate the y\u2019s by multiplying the first equation by 2. Multiply both sides of 2x + y = 7 by 2.\u201d It also shows the steps with equations. Initially the equations are ex + y = 7 and x \u2013 2y = 6. Then they become 2(2x + y) = 2 times 7 and x \u2013 2y = 6. They then become 4x + 2y = 14 and x \u2013 2y = 6.\" data-media-type=\"image\/jpeg\" \/><\/span><span data-type=\"media\" data-alt=\"The third row says, \u201cStep 3: Add the equations resulting from step 2 to eliminate one variable.\u201d It also says, \u201cWe add the x\u2019s, y\u2019s, and constants.\u201d It then gives the equation as 5x = 20.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004c_img_new.jpg\" alt=\"The third row says, \u201cStep 3: Add the equations resulting from step 2 to eliminate one variable.\u201d It also says, \u201cWe add the x\u2019s, y\u2019s, and constants.\u201d It then gives the equation as 5x = 20.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345511013\" data-type=\"media\" data-alt=\"The fourth row says, \u201cStep 4: Solve for the remaining variable.\u201d It also says, \u201cSolve for x.\u201d It gives the equation as x = 4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004d_img_new.jpg\" alt=\"The fourth row says, \u201cStep 4: Solve for the remaining variable.\u201d It also says, \u201cSolve for x.\u201d It gives the equation as x = 4.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345252882\" data-type=\"media\" data-alt=\"The fifth row says, \u201cStep 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.\u201d It also says, \u201cSubstitute x = 4 into the second equation, x \u2013 2y = 6. Then solve for y.\u201d It then gives the equations as x \u2013 2y = 6 which becomes 4 \u2013 2y = 6. This is then \u22122y = 2, and thus, y = \u22121.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004e_img_new.jpg\" alt=\"The fifth row says, \u201cStep 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.\u201d It also says, \u201cSubstitute x = 4 into the second equation, x \u2013 2y = 6. Then solve for y.\u201d It then gives the equations as x \u2013 2y = 6 which becomes 4 \u2013 2y = 6. This is then \u22122y = 2, and thus, y = \u22121.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345436405\" data-type=\"media\" data-alt=\"The sixth row says, \u201cStep 6: Write the solution as an order pair.\u201d It also says, \u201cWrite it as (x, y).\u201d It gives the ordered pair as (4, \u22121).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004f_img_new.jpg\" alt=\"The sixth row says, \u201cStep 6: Write the solution as an order pair.\u201d It also says, \u201cWrite it as (x, y).\u201d It gives the ordered pair as (4, \u22121).\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345406365\" data-type=\"media\" data-alt=\"The seventh row says, \u201cStep 7: Check that the ordered pair is a solution to both original equations.\u201d It also says, \u201cSubstitute (4, \u22121) into 2x + y = 7 and x \u2013 2y = 6. Do they make both equations true? Yes!\u201d It then gives the equations. 2x + y = 7 becomes 2 times 4 + \u22121 = 7 which is 7 = 7. x \u2013 2y = 6 becomes 4 \u2013 2 times \u22121 = 6 which is 6 = 6. The row then says, \u201cThe solution is (4, \u22121).\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_004g_img_new.jpg\" alt=\"The seventh row says, \u201cStep 7: Check that the ordered pair is a solution to both original equations.\u201d It also says, \u201cSubstitute (4, \u22121) into 2x + y = 7 and x \u2013 2y = 6. Do they make both equations true? Yes!\u201d It then gives the equations. 2x + y = 7 becomes 2 times 4 + \u22121 = 7 which is 7 = 7. x \u2013 2y = 6 becomes 4 \u2013 2 times \u22121 = 6 which is 6 = 6. The row then says, \u201cThe solution is (4, \u22121).\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-id1168345633560\" data-type=\"problem\">\n<p id=\"fs-id1168345197716\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d4f5fd209819a980d49402e646c8672f_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;&#45;&#51;&#121;&#61;&#55;&#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-id1168345427288\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345251140\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-25c4864d7eaa7ae5b2fe81ae29cf46af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>The steps are listed below for easy reference.<\/p>\n<div id=\"fs-id1168345256923\" 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 solve a system of equations by elimination.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169752826769\" class=\"stepwise\" type=\"1\">\n<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n<li>Make the coefficients of one variable opposites.\n<ul id=\"fs-id1168345708283\" data-bullet-style=\"bullet\">\n<li>Decide which variable you will eliminate.<\/li>\n<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<\/ul>\n<\/li>\n<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute the solution from Step 4 into one of the original equations. Then solve for 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>\n<\/div>\n<p id=\"fs-id1168345677482\">First we\u2019ll do an example where we can eliminate one variable right away.<\/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-id1168345407940\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345429700\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-775de3b40a845125f2f66e88c0b743b8_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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345230664\" 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-id1167824734317\" style=\"height: 305px; width: 100%;\" summary=\"This figure beings with two equations: x + y = 10 and x \u2013 y = 12. The figure then says, \u201cBoth equations are in standard form. The coefficients of y are already opposites. Add the two equations to eliminate y. The resulting equations has only 1 variable x. It then shows the equations added together. Thus x + y = 10 plus x \u2013 y = 12 becomes 2x = 22. The figure then instructs, \u201cSolve for x, the remaining variable.\u201d Thus x = 11. The figure then says, \u201cSolve for x = 11 into one of the original equations. Thus x + y = 10 becomes 11 + y = 10. The figure then says, \u201cSolve for the other variable, y.\u201d Thus y = -1. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (11, -1).\u201d It then states, \u201cCheck the ordered pair is a solution to both original equations. Thus x + y = 10 becomes 11 + (-1) = 10 and 10 = 10. x \u2013 y = 12 becomes 11 \u2013 (-1) = 12 and 12 = 12. The figure then states, \u201cThe solution is (11, -1).\" data-label=\"\">\n<tbody>\n<tr style=\"height: 19px;\">\n<td style=\"height: 19px; width: 50.8911%;\"><\/td>\n<td style=\"height: 19px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836530489\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">Both equations are in standard form.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">The coefficients of\u00a0<em data-effect=\"italics\">y<\/em>\u00a0are already opposites.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\" data-valign=\"top\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/>\n<\/span>The resulting equation has only 1 variable,\u00a0<em data-effect=\"italics\">x<\/em>.<\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836317852\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px; width: 50.8911%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>, the remaining variable.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 11 into one of the original equations.<\/td>\n<td style=\"height: 46px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167824590565\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\"><\/td>\n<td style=\"height: 14px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167829742244\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 50.8911%;\" data-valign=\"top\">Solve for the other variable,\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"height: 14px; width: 48.9109%;\" data-valign=\"top\"><span id=\"fs-id1167836787837\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_005e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\">The ordered pair is (11, \u22121).<\/td>\n<\/tr>\n<tr style=\"height: 94px;\">\n<td style=\"height: 94px; width: 50.8911%;\" data-valign=\"top\">Check that the ordered pair is a solution<span data-type=\"newline\"><br \/>\n<\/span>to\u00a0<strong data-effect=\"bold\">both<\/strong>\u00a0original 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-09526b5f1a86d524a6b5a813f8e11633_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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#49;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#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=\"63\" width=\"346\" style=\"vertical-align: -26px;\" \/><\/td>\n<td style=\"height: 94px; width: 48.9109%;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 50.8911%;\"><\/td>\n<td style=\"height: 30px; width: 48.9109%;\" data-valign=\"top\">The solution is (11, \u22121).<\/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-id1168345291597\" data-type=\"problem\">\n<p id=\"fs-id1168345277703\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-729c5def8ba39d1a62119c798a1f22e2_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;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345255731\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345427021\"><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;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341961050\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345544062\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f1dfd340bf3c21cf265188aef8cec33c_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;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#45;&#54;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345742599\" 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-id1167836360149\" style=\"width: 100%;\" summary=\"This figure begins with two equations: 3x -2y = -2 and 5x \u2013 6y = 10. The figure instructs, \u201cBoth equations are in standard form. None of the coefficients are opposites. We can make the coefficients of y opposites by multiplying the first equation by -3.\u201d The figure then shows the equations. The first is -3 times (3x \u2013 2y) = -3 times -2, and the second is 5x \u2013 6y = 10. The figure then instructs, \u201cSimplify.\u201d The two equations are -9x + 6y = 6 and 5x \u2013 6y = 10. The figure then says, \u201cAdd the two equations in eliminate y.\u201d The two equations added together becomes -4x = 16. The figure then says, \u201cSolve for the remaining variable x.\u201d Thus x = -4. The figure then instructs, \u201cSubstitute x = -4 into one of the original equations. Thus 3x \u2013 2y = -2 becomes 3 times -4 \u2013 2y = -2. The figure then instructs, \u201cSolve for y.\u201d The equation becomes -12 - 2y = 2 or -2y = 10. Thus y = -5. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (-4, -5).\u201d The figure then says, \u201cCheck that the ordered pair is a solution to both original equations.\u201d Thus 3x -2y = -2 becomes 3 times -4 - 2 times -5 = -2 or -12 +10 = -2 or -2y = -2. It also shows that 5x \u2013 6y = 10 becomes 3 times -4 \u2013 6 times -5 = 10 or -20 + 30 = 10. Thus 10 = 10. The figure then says, \u2018The solutions is (-4, -5).\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 508.047px;\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836609520\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Both equations are in standard form.<\/td>\n<td style=\"width: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">None of the coefficients are opposites.<\/td>\n<td style=\"width: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">We can make the coefficients of\u00a0<em data-effect=\"italics\">y<\/em>\u00a0opposites by multiplying<span data-type=\"newline\"><br \/>\n<\/span>the first equation by \u22123.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167833316764\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167833057104\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836596232\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Solve for the remaining variable,\u00a0<em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\"><br \/>\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= \u22124 into one of the original equations.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167836662540\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167829597819\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\"><span id=\"fs-id1167829619693\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167833274679\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1167836536250\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_006i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\">The ordered pair is (\u22124, \u22125).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\" data-valign=\"top\">Check that the ordered pair is a solution to<span data-type=\"newline\"><br \/>\n<\/span>both original 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-8da8e21610d9c7df368c32f88467e9aa_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;&#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;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#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;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#50;&#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;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#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;&#53;&#120;&#45;&#54;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#48;&#43;&#51;&#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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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=\"93\" width=\"438\" style=\"vertical-align: -42px;\" \/><\/td>\n<td style=\"width: 342.172px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 508.047px;\"><\/td>\n<td style=\"width: 342.172px;\" data-valign=\"top\">The solution is (\u22124, \u22125).<\/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 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p>Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8e17fb9f9252c179c750e6db1df4f176_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#45;&#57;&#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;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345212163\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345734996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-745bd1150b5c3e65ae8bad5282a5b3b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345507881\">Now we\u2019ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.<\/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-id1168345556370\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345744930\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9f22a31efb5a926b4e94a9739fd9cf7f_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;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#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;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345284539\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p>In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by a constant to get the opposites.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167836530249\" style=\"width: 100%;\" summary=\"This figure begins with two equations: 4x \u2013 3y = 9 and 7x + 2y = -6. The figure then says, \u201cBoth equations are in standard form. To get opposite coefficients of y, we will multiply the first equation by 2 and the second equation by 3.\u201d It then shows the equations as 2 times (4x \u2013 3y) = 2 times 9 and 3 times (7x + 2y) = 3 times -6. The figure then says, \u201cSimplify.\u201d The equations then become 8x \u2013 6y = 18 and 21x + 6y = -18. The figure then says, \u201cAdd the two equations to eliminate y. After adding, the answer is 39x = 0. The figure then says, \u201cSolve for x.\u201d Thus, x = 0. The figure then reads, \u201cSubstitute x = 0 into one of the original equations.\u201d Thus 7x +2y = -6 becomes 7 times 0 + 2y = -6. The figure then says, \u201cSolve for y.\u201d It then says, 2y = -6 and thus 2y = -3. The figure then reads, \u201cWrite the solution as an ordered pair. The ordered pair is (0, -3).\u201d The figure then instructs, \u201cCheck that the ordered pair is a solution to both original equations. Thus 4x \u2013 3y = 9 becomes 4 times 0 \u2013 3 times -3 = 9 or 9 = 9. Thus 7x + 2y = -6 becomes 7 times 0 + 2 times -3 = -6 or -6 = -6. The figure then says, \u201cThe solution is (0, -3).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836516310\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Both equations are in standard form. To get opposite<span data-type=\"newline\"><br \/>\n<\/span>coefficients of\u00a0<em data-effect=\"italics\">y<\/em>, we will multiply the first equation by 2<span data-type=\"newline\"><br \/>\n<\/span>and the second equation by 3.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167829693355\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167829850392\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"middle\">Add the two equations to eliminate\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167832930182\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">x<\/em>.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 0 into one of the original equations.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836349138\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836688612\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836306660\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\"><span id=\"fs-id1167836503981\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_007h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\">The ordered pair is (0, \u22123).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\" data-valign=\"top\">Check that the ordered pair is a solution to<span data-type=\"newline\"><br \/>\n<\/span><strong data-effect=\"bold\">both<\/strong>\u00a0original 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-0f84d78ab06c02dead7c4167458a4d61_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;&#45;&#51;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#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;&#55;&#120;&#43;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"401\" style=\"vertical-align: -25px;\" \/><\/td>\n<td style=\"width: 52.4752%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.3267%;\"><\/td>\n<td style=\"width: 52.4752%;\" data-valign=\"top\">The solution is (0, \u22123).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345367998\">\n<\/div>\n<\/div>\n<\/div>\n<p>What other constants could we have chosen to eliminate one of the variables? Would the solution be the same?<\/p>\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-id1168345406843\" data-type=\"problem\">\n<p id=\"fs-id1168345360570\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-55873f5fc44c4fcf48e1845cfe6825f2_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;&#52;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345414924\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345511147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-450ee1faefebebc715b20a97daae94ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345419206\">When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by its LCD.<\/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-id1168341960861\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345292398\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a3f00986f5a05aa73e09f2607e6dfa4_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#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=\"44\" width=\"122\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345404754\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1168345217762\">In this example, both equations have fractions. Our first step will be to multiply each equation by its LCD to clear the fractions.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167836409014\" class=\"unnumbered unstyled can-break\" summary=\"This figure beings with two equations: x + (1\/2)y = 6 and (3\/2)x + (2\/3)y = 17\/2. The figure says, \u201cTo clear the fractions multiply each equation by its LCD.\u201d It then shows the equations as 2 times (x + (1\/2)y) = 2 times 6 and 6 times ((3\/2)x + (2\/3)y) = 6 times (17\/2). The figure then says, \u201cSimplify.\u201d The equations then become 2x + y = 12 and 9x + 4y = 51. The figure then says, \u201cNow we are ready to eliminate one of the variables. Notice that both equations are in standard form. We can eliminate y multiplying the top equation by -4.\u201d It then shows -4 times (2x + y) = -4 times 12 and 9x + 4y = 51. The figure then says, \u201cSimplify and add.\u201d The equations added are thus -8x \u2013 4y = -48 plus 9x + 4y = 51 which gives x = 3. The figure then says, \u201cSubstitute x = 3 into one of the original equations. Solve for y.\u201d Thus x + (1\/2)y = 6 becomes 3 + (1\/2)y = 6. This becomes (1\/2)y = 3 or y = 6. The figure then says, \u201cWrite the solution as an ordered pair. The ordered pair is (3, 6). The figure then says, \u201cCheck the ordered pair is a solution to both original equations. Thus x + (1\/2)y = 6 becomes 3 + (1\/2) times 6 = 6 or 3 + 6 = 6. Thus 6 = 6. The second equation is (3\/2)x + (2\/3)y = 17\/2 or (3\/2) times 3 + (2\/3) times 6 = 17\/2. This becomes 9\/2 + 4 = 17\/2 or 9\/2 + 8\/2 = 17\/2. Thus 17\/2 = 17\/2. The figure then says, \u201cThe solution is (3, 6).\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836688835\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">To clear the fractions, multiply each equation by its LCD.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836502141\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Simplify.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836352044\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Now we are ready to eliminate one of the variables. Notice that<span data-type=\"newline\"><br \/>\n<\/span>both equations are in standard form.<\/td>\n<td style=\"width: 289px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">We can eliminate\u00a0<em data-effect=\"italics\">y<\/em>\u00a0multiplying the top equation by \u22124.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836560117\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Simplify and add.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Substitute\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 into one of the original equations.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836300298\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">y<\/em>.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836342400\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836388312\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\"><span id=\"fs-id1167836627560\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_008h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Write the solution as an ordered pair.<\/td>\n<td style=\"width: 289px;\" data-valign=\"top\">The ordered pair is (3, 6).<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\" data-valign=\"top\">Check that the ordered pair is a solution<span data-type=\"newline\"><br \/>\n<\/span>to\u00a0<strong data-effect=\"bold\">both<\/strong>\u00a0original 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-a3d7681a420b2d8f63f1db072d97cb90_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#43;&#54;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#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;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#43;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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=\"156\" width=\"347\" style=\"vertical-align: -58px;\" \/><\/td>\n<td style=\"width: 289px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 185px;\"><\/td>\n<td style=\"width: 289px;\" data-valign=\"top\">The solution is (3, 6).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345428888\" data-type=\"problem\">\n<p>Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c07f8d0af2b6176c1a17c1d0fcaf678b_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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=\"44\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345286997\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345290805\"><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-id1168345357619\">When we were solving systems of linear equations by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.<\/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-id1168345287937\" data-type=\"note\">\n<div id=\"fs-id1168345213647\" data-type=\"exercise\">\n<div id=\"fs-id1168345645051\" data-type=\"problem\">\n<p id=\"fs-id1168345550525\">Solve the system by elimination:<\/p>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75d12a3c477aa3db4d1ce4809da79bd6_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;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-408902b575807d54ea482653068f6f35_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;&#51;&#121;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#53;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>c) <span style=\"text-align: initial; font-size: 0.9em; word-spacing: normal;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95cc90d48d7480be1d320abe6e2e3660_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;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#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=\"118\" style=\"vertical-align: -17px;\" \/><\/span><\/p>\n<p>d)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9dbf8a6565cd0beff8e45730c800d442_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;&#54;&#120;&#43;&#49;&#53;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#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=\"143\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1168345346541\" data-type=\"solution\">\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-149\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 410.094px;\">a)<\/td>\n<td style=\"width: 440.125px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75d12a3c477aa3db4d1ce4809da79bd6_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;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Write the second equation in standard form.<\/td>\n<td style=\"width: 440.125px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2cd91f807daf5ba09a56a53b7405da6_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#43;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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=\"141\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Clear the fractions by multiplying the second equation by 4.<\/td>\n<td style=\"width: 440.125px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4a976c1470fa0ae8a902a8806d272a6_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#43;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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=\"179\" style=\"vertical-align: -18px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">Simplify.<\/td>\n<td style=\"width: 440.125px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f7d249cdde88cb4ec84c178e7e8613a4_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#43;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#43;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"141\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">To eliminate a variable, we multiply the second equation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>.\u00a0<span data-type=\"newline\"><br \/>\n<\/span>Simplify and add.<\/td>\n<td style=\"width: 440.125px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8dcd83352a6157145ba0bbf94738ea4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#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;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#120;&#45;&#52;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#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=\"59\" width=\"169\" style=\"vertical-align: -22px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">This is a true statement. The equations are consistent but dependent. Their graphs would be the same line. The system has infinitely many solutions.<\/td>\n<td style=\"width: 440.125px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 410.094px;\">After we cleared the fractions in the second equation, did you notice that the two equations were the same? That means we have coincident lines.<\/td>\n<td style=\"width: 440.125px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">b)<\/td>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1168345213649\" data-type=\"problem\">\n<p id=\"fs-id1168345539011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-408902b575807d54ea482653068f6f35_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;&#51;&#121;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#53;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168341917062\" data-type=\"solution\">\n<p id=\"fs-id1168345415533\">\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">infinitely many solutions<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1168345213649\" data-type=\"problem\"><\/div>\n<div id=\"fs-id1168341917062\" data-type=\"solution\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">c)<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95cc90d48d7480be1d320abe6e2e3660_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;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#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=\"118\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">infinitely many solutions<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341961563\" data-type=\"note\">\n<div data-type=\"exercise\">\n<div id=\"fs-id1168345511489\" data-type=\"problem\"><\/div>\n<div id=\"fs-id1168345450687\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<table id=\"eip-353\" class=\"unnumbered unstyled\" style=\"height: 192px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 429px;\">d)<\/td>\n<td style=\"height: 16px; width: 420px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9dbf8a6565cd0beff8e45730c800d442_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;&#54;&#120;&#43;&#49;&#53;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#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=\"143\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 429px;\">The equations are in standard form.<\/td>\n<td style=\"height: 32px; width: 420px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c17e7e55adac49cce060aa57d16861ab_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#120;&#43;&#49;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#45;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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=\"169\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 429px;\">Multiply the second equation by 3 to eliminate a variable.<\/td>\n<td style=\"height: 32px; width: 420px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c875f25d1d234cbb42b4d3729b0666b9_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#120;&#43;&#49;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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=\"198\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 429px;\">Simplify and add.<\/td>\n<td style=\"height: 48px; width: 420px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4507b29ade972d56ce5d643281c6d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#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;&#45;&#54;&#120;&#43;&#49;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#120;&#45;&#49;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#110;&#101;&#32;&#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=\"63\" width=\"178\" style=\"vertical-align: -26px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 429px;\">This statement is false. The equations are inconsistent and so their graphs would be parallel lines.<\/td>\n<td style=\"height: 48px; width: 420px;\"><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 429px;\">The system does not have a solution.<\/td>\n<td style=\"height: 16px; width: 420px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341952290\" data-type=\"problem\">\n<p id=\"fs-id1168341952292\">Solve the system by elimination. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ef0803dc8da7be9b98602f8eccb32ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#50;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#57;&#120;&#45;&#54;&#121;&#61;&#49;&#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;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345447658\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345447660\">no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168341916882\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Equations by Elimination<\/h1>\n<p id=\"fs-id1168341916887\">Some applications problems translate directly into equations in standard form, so we will use the elimination method to solve them. As before, we use our Problem Solving Strategy to help us stay focused and organized.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345255373\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p>The sum of two numbers is 39. Their difference is 9. Find the numbers.<\/p>\n<\/div>\n<div 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=\"eip-269\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td style=\"width: 524.562px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 524.562px;\">We are looking for two numbers.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 3. Name<\/strong>\u00a0what we are looking for.\u00a0<span data-type=\"newline\"><br \/>\n<\/span>Choose a variable to represent that quantity.<\/td>\n<td style=\"width: 524.562px;\">Let <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.\u00a0<span data-type=\"newline\"><br \/>\n<\/span><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 style=\"width: 325.656px;\"><strong>Step 4. Translate<\/strong>\u00a0into a system of equations.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>The system is:<\/td>\n<td style=\"width: 524.562px;\">The sum of two numbers is 39.\u00a0<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d9d1f436631bbb1ac0cf53769972c789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#43;&#109;&#61;&#51;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -2px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>Their difference is 9.\u00a0<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0bcce422bdb41c3df3040624e7e615b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#45;&#109;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#110;&#43;&#109;&#61;&#51;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#45;&#109;&#61;&#57;&#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=\"62\" width=\"109\" style=\"vertical-align: -28px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 5. Solve<\/strong>\u00a0the system of equations.\u00a0<span data-type=\"newline\"><br \/>\n<\/span>To solve the system of equations, use elimination.\u00a0<span data-type=\"newline\"><br \/>\n<\/span>The equations are in standard form and the coefficients of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> are opposites. Add.\u00a0<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>.\u00a0<span data-type=\"newline\"><br \/>\n<\/span><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-cde34f8f41c9db6d7bf5f5ae25d07167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> into one of the original equations and solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td style=\"width: 524.562px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d69097ab04487e95ec1db83eaed6f9cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#110;&#43;&#109;&#61;&#51;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#45;&#109;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#52;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#50;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#43;&#109;&#61;&#51;&#57;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#52;&#43;&#109;&#61;&#51;&#57;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#61;&#49;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"169\" width=\"119\" style=\"vertical-align: -77px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td style=\"width: 524.562px;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6172f88ba06a0dc4a7a67703117dea41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#43;&#49;&#53;&#61;&#51;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"99\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c66dd68119d802706110fee96a5ad12d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#49;&#53;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -1px;\" \/>, the answers check.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 325.656px;\"><strong>Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td style=\"width: 524.562px;\">The numbers are 24 and 15.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345577233\" data-type=\"problem\">\n<p id=\"fs-id1168345577235\">The sum of two numbers is 42. Their difference is 8. Find the numbers.<\/p>\n<\/div>\n<div id=\"fs-id1168345577239\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345577241\">The numbers are 25 and 17.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\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-id1168345542428\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168345542430\">Joe stops at a burger restaurant every day on his way to work. Monday he had one order of medium fries and two small sodas, which had a total of 620 calories. Tuesday he had two orders of medium fries and one small soda, for a total of 820 calories. How many calories are there in one order of medium fries? How many calories in one small soda?<\/p>\n<\/div>\n<div id=\"fs-id1168345665125\" 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-id1167836534642\" style=\"width: 100%;\" summary=\"This figure instructs, \u201cStep 1. Read the problem. Step 2. Identify what we are looking for. We are looking for the number of calories in one order of medium fires and in one small soda. Step 3. Name what we are looking for. Let f = the number of calories in 1 order of medium fries. s = the number of calories in 1 small soda. Step 4. Translate into a system of equations: one medium fries and two small sodas had a total of 620 calories. f + 2s = 620. Two medium fries and one small soda had a total of 820 calories. 2f + s = 820. Our syste is f + 2s = 620 and 2f + s = 820. Step 5. Solve the system of equations. To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of f, multiply the top equation by -2.\u201d The equations are -2(f + 2s) = -2 times 620 and 2f + s =820. The figure then says, \u201cSimplify and add.\u201d Thus -2f \u2013 4s = -1240 plus 2f + s = 820 equals -3s = -420. The figure then says, \u201cSolve for s.\u201d Thus s = 140. The figure then reads, \u201cSubstitute s = 140 into one of the original equations and then solve for f. Thus, f + 2s = 620 becomes f + 2 times 140 = 620 or f +280 = 620. Thus f = 340. The figure then reads, \u201cStep 6. Check the answer. Verify that these numbers make sense in the problem and that they are solutions to both equations. We leave this to you! Step 7. Answer the question. The small soda has 140 calories and the fries have 340 calories.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 1. Read<\/strong>\u00a0the problem.<\/td>\n<td style=\"width: 57.2727%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 2. Identify<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">We are looking for the number of<span data-type=\"newline\"><br \/>\n<\/span>calories in one order of medium fries<span data-type=\"newline\"><br \/>\n<\/span>and in one small soda.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 3. Name<\/strong>\u00a0what we are looking for.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">Let\u00a0<em data-effect=\"italics\">f<\/em>\u00a0= the number of calories in<span data-type=\"newline\"><br \/>\n<\/span>1 order of medium fries.<span data-type=\"newline\"><br \/>\n<\/span>\u2003\u00a0\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= the number of calories in<span data-type=\"newline\"><br \/>\n<\/span>1 small soda.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 4. Translate<\/strong>\u00a0into a system of equations:<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">one medium fries and two small sodas had a<span data-type=\"newline\"><br \/>\n<\/span>total of 620 calories<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836492134\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">two medium fries and one small soda had a<span data-type=\"newline\"><br \/>\n<\/span>total of 820 calories.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167824658652\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Our system is:<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836319166\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 5. Solve<\/strong>\u00a0the system of equations.<span data-type=\"newline\"><br \/>\n<\/span>To solve the system of equations, use<span data-type=\"newline\"><br \/>\n<\/span>elimination. The equations are in standard<span data-type=\"newline\"><br \/>\n<\/span>form. To get opposite coefficients of\u00a0<em data-effect=\"italics\">f<\/em>,<span data-type=\"newline\"><br \/>\n<\/span>multiply the top equation by \u22122.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836406845\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Simplify and add.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167833356376\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009e_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Solve for\u00a0<em data-effect=\"italics\">s<\/em>.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167833338872\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009f_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\">Substitute\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 140 into one of the original<span data-type=\"newline\"><br \/>\n<\/span>equations and then solve for\u00a0<em data-effect=\"italics\">f<\/em>.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836318741\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009g_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836533791\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009h_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836507422\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009i_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\"><\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\"><span id=\"fs-id1167836549057\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_03_009j_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 6. Check<\/strong>\u00a0the answer.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">Verify that these numbers make sense<span data-type=\"newline\"><br \/>\n<\/span>in the problem and that they are<span data-type=\"newline\"><br \/>\n<\/span>solutions to both equations.<span data-type=\"newline\"><br \/>\n<\/span>We leave this to you!<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.6136%;\" data-valign=\"top\"><strong data-effect=\"bold\">Step 7. Answer<\/strong>\u00a0the question.<\/td>\n<td style=\"width: 57.2727%;\" data-valign=\"top\">The small soda has 140 calories and<span data-type=\"newline\"><br \/>\n<\/span>the fries have 340 calories.<\/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 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341862434\" data-type=\"problem\">\n<p id=\"fs-id1168341862436\">Malik stops at the grocery store to buy a bag of diapers and 2 cans of formula. He spends a total of &#36;37. The next week he stops and buys 2 bags of diapers and 5 cans of formula for a total of &#36;87. How much does a bag of diapers cost? How much is one can of formula?<\/p>\n<\/div>\n<div id=\"fs-id1168345229866\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345229868\">The bag of diapers costs ?11 and the can of formula costs ?13.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345416871\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/h1>\n<p id=\"fs-id1168345550343\">When you will have to solve a system of linear equations in a later math class, you will usually not be told which method to use. You will need to make that decision yourself. So you\u2019ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.<\/p>\n<p><span id=\"fs-id1169752961745\" data-type=\"media\" data-alt=\"This table has two rows and three columns. The first row labels the columns as \u201cGraphing,\u201d \u201cSubstitution,\u201d and \u201cElimination.\u201d Under \u201cGraphing\u201d it says, \u201cUse when you need a picture of the situation.\u201d Under \u201cSubstitution\u201d it says, \u201cUse when one equation is already solved for one variable.\u201d Under \u201cElimination\u201d it says, \u201cUse when the equations are in standard form.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_01_021_img.jpg\" alt=\"This table has two rows and three columns. The first row labels the columns as \u201cGraphing,\u201d \u201cSubstitution,\u201d and \u201cElimination.\u201d Under \u201cGraphing\u201d it says, \u201cUse when you need a picture of the situation.\u201d Under \u201cSubstitution\u201d it says, \u201cUse when one equation is already solved for one variable.\u201d Under \u201cElimination\u201d it says, \u201cUse when the equations are in standard form.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/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-id1168341863749\" style=\"font-size: 16.8px;\" data-type=\"problem\">\n<p id=\"fs-id1168341863751\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1168341863756\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2746097fc625dc40233a8104017b121_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;&#56;&#121;&#61;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#45;&#52;&#121;&#61;&#45;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dedf6d9e8b4daa707f09e1f73d9008f0_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;&#43;&#54;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><strong style=\"font-size: 16.8px; word-spacing: normal;\">Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1168341923301\" style=\"font-size: 16.8px;\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1163872390948\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a4a1652554f79b590bca1c9d74cc07c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#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;&#56;&#121;&#61;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#45;&#52;&#121;&#61;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>Since both equations are in standard form, using elimination will be most convenient.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span class=\"token\">b)<\/span>\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7fdfeb53347c8d544cdb8e7c13720ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#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;&#43;&#54;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#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=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>Since one equation is already solved for\u00a0<em data-effect=\"italics\">y<\/em>, using substitution will be most convenient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345580362\" data-type=\"problem\">\n<p id=\"fs-id1168345580364\">For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1168345580369\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-105f0dd9eea0089e7d13f1596a1b3a61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#53;&#121;&#61;&#45;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b1313a3247883c741a09911c6033a596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#50;&#121;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#53;&#121;&#61;&#45;&#55;&#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-id1168345443554\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345443556\">a) Since both equations are in standard form, using elimination will be most convenient.<\/p>\n<p>b) Since one equation is already solved 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;\" \/>, using substitution will be most convenient.<\/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 linear equations by elimination.<\/span><\/p>\n<div class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1168345665200\" data-display=\"block\">\n<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination1\">Instructional Video-Solving Systems of Equations by Elimination<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination2\">Instructional Video-Solving by Elimination<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25Elimination3\">Instructional Video-Solving Systems by Elimination<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168345723908\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">To Solve a System of Equations by Elimination<\/strong>\n<ol id=\"fs-id1169750655024\" class=\"stepwise\" type=\"1\">\n<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n<li>Make the coefficients of one variable opposites.\n<ul id=\"fs-id1171790125566\" data-bullet-style=\"circled\">\n<li>Decide which variable you will eliminate.<\/li>\n<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<\/ul>\n<\/li>\n<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute the solution from Step 4 into one of the original equations. Then solve for 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<\/li>\n<\/ul>\n<div id=\"fs-id1168341962936\" class=\"practice-perfect\" data-depth=\"2\">\n<h1 data-type=\"title\">4.3 Exercise Set<\/h1>\n<p id=\"fs-id1169747646746\">In the following exercises, solve the systems of equations by elimination.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6ee8ec8e7e6296491a829b44fc373810_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d32301dce20ac2e9304cd2b4eada5e61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#121;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#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-93741e4f63bfb62463e5a062611b8579_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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#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=\"107\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c0c99350b84013112f681998b6edfb2a_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;&#55;&#120;&#43;&#54;&#121;&#61;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#54;&#121;&#61;&#50;&#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=\"148\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-06b13791c5e72bb76a5fd79973e384ca_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;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#53;&#120;&#45;&#52;&#121;&#61;&#45;&#55;&#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-eeeda31e5bc2050eb81822f90a595dfb_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;&#52;&#121;&#61;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9a8371e110e1ca54bc0211985bc59e51_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;&#54;&#120;&#45;&#53;&#121;&#61;&#45;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#45;&#50;&#121;&#61;&#45;&#49;&#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=\"139\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bc3e2d1d46c0a0ade0b9492ea2c0dd2d_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;&#53;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#61;&#49;&#55;&#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-9c782e7c46638b5e76b50371c9c6553c_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;&#55;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#51;&#120;&#43;&#51;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-135991749befb3e82b1307c43199bf94_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;&#53;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#50;&#121;&#61;&#49;&#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-b75d693aef038fd37d0c33db54dae2e7_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;&#55;&#121;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2ccd8d1a3f8aa7e63619e7075daf6d0a_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;&#56;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7da54d02f8290ddcaa21b4630f851688_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;&#56;&#121;&#61;&#54;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#54;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3d2e9ace77e33442a931fb48fe1df16f_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" 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-f745a9fe3359b90b446bd41b41d840c8_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-80c4731a316ba331082ef53183022132_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;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf7e284daf756541b93485cdc1b7c8d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#45;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#50;&#121;&#61;&#45;&#49;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c0abb18c3166037d4e34b82efa5555d1_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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#54;&#120;&#45;&#52;&#121;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"147\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-554f1cedbd86b743227672c2945a121d_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;&#49;&#49;&#120;&#43;&#49;&#50;&#121;&#61;&#54;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#50;&#120;&#43;&#50;&#52;&#121;&#61;&#57;&#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=\"151\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9ebbb37b88c17d74490986382078fe1c_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;&#51;&#121;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1169747673556\">In the following exercises, translate to a system of equations and solve.<\/p>\n<ol start=\"21\">\n<li>The sum of two numbers is 65. Their difference is 25. Find the numbers<span style=\"text-align: initial; background-color: initial; font-size: 0.9em;\">.<\/span><\/li>\n<li>The sum of two numbers is \u221227. Their difference is \u221259. Find the numbers.<\/li>\n<li>Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for &#36;114 or she is able to buy 2 shirts and 4 sweaters for &#36;164. How much does a shirt cost? How much does a sweater cost?<\/li>\n<li>The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?<\/li>\n<\/ol>\n<p id=\"fs-id1169752961726\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p>\n<ol start=\"25\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0fbcf17fee700e3df2b21aa0979d18f5_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;&#56;&#120;&#45;&#49;&#53;&#121;&#61;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#51;&#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=\"142\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2f7b5a2c671c59d629bdc2113be4ed0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#52;&#121;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5711d3fe2c1abce97492d5f1a56d87_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;&#52;&#120;&#43;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#45;&#50;&#121;&#61;&#45;&#50;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3da1863cc4b90390bdcb246fde9876e_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;&#57;&#120;&#45;&#52;&#121;&#61;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#53;&#121;&#61;&#45;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>Norris can row 3 miles upstream against the current in the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-610e64c2fa2d8015b7a21952be71b117_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;&#114;&#45;&#99;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#114;&#43;&#99;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"90\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li>for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, his rowing speed in still water.<\/li>\n<li>Then solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, the speed of the river current.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1168345487655\" data-type=\"exercise\">\n<div id=\"fs-id1168345487657\" data-type=\"problem\">\n<p id=\"fs-id1168345487659\"><span data-type=\"newline\"><span style=\"background-color: initial; font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Answers:<\/span><br \/>\n<\/span><\/p>\n<ol class=\"twocolumn\">\n<li>(6, 9)<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-59362f5af96d469003752c4e94766b84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ecc91b8d5e91ea23c08fcf2ee52342c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f2347704000ad2e9ae878a8427611b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aeaca1910c48ce1d63188a726ff94fc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#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-a32114edb1329de201f25f57cba4ea20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>(6, 1)<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7398c7acd0cb4db6c7a69dee48be09ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>(2, 3)<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f5daa582fe5aaa911a134d0bfd3cfe1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-403bc0cb12c4fb4e2bf022eff0167913_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>(9, 5)<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f307aa6b65fb69739e6c2b7458409de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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-6fd7f677a681964debbd5fb9bbb3c944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>infinitely many solutions<\/li>\n<li>infinitely many solutions<\/li>\n<li>infinitely many solutions<\/li>\n<li>inconsistent, no solution<\/li>\n<li>inconsistent, no solution<\/li>\n<li>The numbers are 20 and 45.<\/li>\n<li>The numbers are 16 and \u221243.<\/li>\n<li>A shirt costs &#36;16 and a sweater costs &#36;33.<\/li>\n<li>There are 860 mg in a hot dog. There are 1,000 mg in a cup of cottage cheese.<\/li>\n<li>\n<ol type=\"a\">\n<li>elimination<\/li>\n<li>substitution<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>substitution<\/li>\n<li>elimination<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b14f8d37554b47394ab123bfa86ff407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8bc0f94c08cf058d316895e763d34082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":125,"menu_order":3,"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-1297","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\/1297","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\/1297\/revisions"}],"predecessor-version":[{"id":1298,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1297\/revisions\/1298"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/1094"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1297\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1297"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1297"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1297"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}