{"id":10674,"date":"2017-06-05T14:58:14","date_gmt":"2017-06-05T14:58:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10674"},"modified":"2018-01-15T18:34:22","modified_gmt":"2018-01-15T18:34:22","slug":"creating-a-table-of-ordered-pair-solutions-to-a-linear-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/creating-a-table-of-ordered-pair-solutions-to-a-linear-equation\/","title":{"raw":"Creating a Table of Ordered Pair Solutions to a Linear Equation","rendered":"Creating a Table of Ordered Pair Solutions to a Linear Equation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Complete a table of values that satisfy a two variable equation<\/li>\r\n \t<li>Find any solution to a two variable equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p data-type=\"title\">In the previous examples, we substituted the [latex]x\\text{- and }y\\text{-values}[\/latex] of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[\/latex] and then solve the equation for [latex]y[\/latex]. Or, choose a value for [latex]y[\/latex] and then solve for [latex]x[\/latex].<\/p>\r\n<p data-type=\"title\">We\u2019ll start by looking at the solutions to the equation [latex]y=5x - 1[\/latex] we found in the previous chapter. We can summarize this information in a table of solutions.<\/p>\r\n\r\n<table id=\"fs-id1801596\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row has the equation y = 5 x -1. The first row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x - 1[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\"><\/td>\r\n<td data-align=\"left\"><\/td>\r\n<td data-align=\"left\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo find a third solution, we\u2019ll let [latex]x=2[\/latex] and solve for [latex]y[\/latex].\r\n<table id=\"eip-id1168469817063\" class=\"unnumbered\" summary=\"The figure shows a substitution into an equation and accompanying comments. The first equation is y = 5 open parentheses 2, shown in blue, closed parentheses - 1. The comment is \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]y=5x - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]x=2[\/latex]<\/td>\r\n<td>[latex]y=5(\\color{blue}{2})-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]y=10 - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe ordered pair is a solution to [latex]y=5x - 1[\/latex]. We will add it to the table.\r\n<table id=\"fs-id1569134\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 5 x - 1. The next row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x - 1[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(2,9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can find more solutions to the equation by substituting any value of [latex]x[\/latex] or any value of [latex]y[\/latex] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nComplete the table to find three solutions to the equation [latex]y=4x - 2\\text{:}[\/latex]\r\n<table id=\"fs-id1599948\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\"><\/td>\r\n<td data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\r\n<td data-align=\"left\"><\/td>\r\n<td data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\"><\/td>\r\n<td data-align=\"left\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSolution\r\nSubstitute [latex]x=0,x=-1[\/latex], and [latex]x=2[\/latex] into [latex]y=4x - 2[\/latex].\r\n<table id=\"eip-id1168468326216\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation. The first substitution x = 0, with 0 shown in blue. The next line is y = 4 x - 2. The next line is y = 4 times 0, shown in blue, minus 2. The next line is y = 0 - 2. The next line is y = -2. The last line is \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-1}[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<td>[latex]y=4x - 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=4\\cdot{\\color{blue}{0}}-2[\/latex]<\/td>\r\n<td>[latex]y=4(\\color{blue}{-1})-2[\/latex]<\/td>\r\n<td>[latex]y=4\\cdot{\\color{blue}{2}}-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=0 - 2[\/latex]<\/td>\r\n<td>[latex]y=-4 - 2[\/latex]<\/td>\r\n<td>[latex]y=8 - 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y=-2[\/latex]<\/td>\r\n<td>[latex]y=-6[\/latex]<\/td>\r\n<td>[latex]y=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe results are summarized in the table.\r\n<table id=\"fs-id1572080\" class=\"unnumbered\" summary=\"This table has 5 rows and three columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(-1,-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(2,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146945[\/ohm_question]\r\n\r\n[ohm_question]146947[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nComplete the table to find three solutions to the equation [latex]5x - 4y=20\\text{:}[\/latex]\r\n<table id=\"fs-id1328205\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table is 5 rows and 3 columns. The first row is the equation 5 x - 4 y = 20. The next row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px;\" colspan=\"3\" data-align=\"center\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px;\" data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px;\" data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"471577\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"471577\"]\r\n\r\nSolution\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224816\/CNX_BMath_Figure_11_01_038_img.png\" alt=\"The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is \" data-media-type=\"image\/png\" \/>\r\nThe results are summarized in the table.\r\n<table id=\"fs-id1572845\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table has 5 rows and 3 columns. The first row is equation 5 x - 4 y = 20. The next row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px;\" colspan=\"3\" data-align=\"center\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px;\" data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px;\" data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(4,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 114px;\" data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(8,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146948[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Find Solutions to Linear Equations in Two Variables<\/h3>\r\nTo find a solution to a linear equation, we can choose any number we want to substitute into the equation for either [latex]x[\/latex] or [latex]y[\/latex]. We could choose [latex]1,100,1,000[\/latex], or any other value we want. But it\u2019s a good idea to choose a number that\u2019s easy to work with. We\u2019ll usually choose [latex]0[\/latex] as one of our values.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind a solution to the equation [latex]3x+2y=6[\/latex]\r\n[reveal-answer q=\"166017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"166017\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469770411\" class=\"unnumbered unstyled\" summary=\"The figure shows a four step solution. Step 1 reads \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"top\"><strong>Step 1:<\/strong> Choose any value for one of the variables in the equation.<\/td>\r\n<td>We can substitute any value we want for [latex]x[\/latex] or any value for [latex]y[\/latex].\r\n\r\nLet's pick [latex]x=0[\/latex].\r\n\r\nWhat is the value of [latex]y[\/latex] if [latex]x=0[\/latex] ?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\"><strong>Step 2:<\/strong> Substitute that value into the equation.\r\n\r\nSolve for the other variable.<\/td>\r\n<td>Substitute [latex]0[\/latex] for [latex]x[\/latex].\r\n\r\nSimplify.\r\n\r\nDivide both sides by [latex]2[\/latex].<\/td>\r\n<td>[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3\\cdot\\color{blue}{0}+2y=6[\/latex]\r\n\r\n[latex]0+2y=6[\/latex]\r\n\r\n[latex]2y=6[\/latex]\r\n\r\n[latex]y=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3:<\/strong> Write the solution as an ordered pair.<\/td>\r\n<td>So, when [latex]x=0,y=3[\/latex].<\/td>\r\n<td>This solution is represented by the ordered pair [latex]\\left(0,3\\right)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\"><strong>Step 4:<\/strong> Check.<\/td>\r\n<td>Substitute [latex]x=\\color{blue}{0}, y=\\color{red}{3}[\/latex] into the equation [latex]3x+2y=6[\/latex]\r\n\r\nIs the result a true equation?\r\n\r\nYes!<\/td>\r\n<td>[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3\\cdot\\color{blue}{0}+2\\cdot\\color{red}{3}\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]0+6\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6=6\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147000[\/ohm_question]\r\n\r\n<\/div>\r\nWe said that linear equations in two variables have infinitely many solutions, and we\u2019ve just found one of them. Let\u2019s find some other solutions to the equation [latex]3x+2y=6[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind three more solutions to the equation [latex]3x+2y=6[\/latex]\r\n[reveal-answer q=\"645203\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645203\"]\r\n\r\nSolution\r\nTo find solutions to [latex]3x+2y=6[\/latex], choose a value for [latex]x[\/latex] or [latex]y[\/latex]. Remember, we can choose any value we want for [latex]x[\/latex] or [latex]y[\/latex]. Here we chose [latex]1[\/latex] for [latex]x[\/latex], and [latex]0[\/latex] and [latex]-3[\/latex] for [latex]y[\/latex].\r\n<table id=\"eip-id1168468473621\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first substitution is y = 0, with 0 shown in red The next line is 3 x + 2 y = 6. The next line is3 x + 2 open parentheses 0, shown in red, closed parentheses = 6. It has the comment \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"bottom\">Substitute it into the equation.<\/td>\r\n<td>[latex]y=\\color{red}{0}[\/latex]\r\n\r\n[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3x+2(\\color{red}{0})=6[\/latex]<\/td>\r\n<td>[latex]y=\\color{blue}{1}[\/latex]\r\n\r\n[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3(\\color{blue}{1})+2y=6[\/latex]<\/td>\r\n<td>[latex]y=\\color{red}{-3}[\/latex]\r\n\r\n[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3x+2(\\color{red}{-3})=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.\r\n\r\nSolve.<\/td>\r\n<td>[latex]3x+0=6[\/latex]\r\n\r\n[latex]3x=6[\/latex]<\/td>\r\n<td>[latex]3+2y=6[\/latex]\r\n\r\n[latex]2y=3[\/latex]<\/td>\r\n<td>[latex]3x-6=6[\/latex]\r\n\r\n[latex]3x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]y=\\Large\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]x=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">Write the ordered pair.<\/td>\r\n<td data-align=\"center\">[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nCheck your answers.\r\n<table id=\"eip-id1168466166098\" class=\"unnumbered unstyled\" summary=\"The figure shows three substitutions into equations. The first starts with \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3\\cdot\\color{blue}{2}+2\\cdot\\color{red}{0}\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6+0\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6+6\\checkmark[\/latex]<\/td>\r\n<td>[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3\\cdot\\color{blue}{1}+2\\cdot\\color{red}{\\Large\\frac{3}{2}}\\normalsize\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]3+3\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6+6\\checkmark[\/latex]<\/td>\r\n<td>[latex]3x+2y=6[\/latex]\r\n\r\n[latex]3\\cdot\\color{blue}{4}+2\\cdot\\color{red}{-3}\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]12+(-6)\\stackrel{?}{=}6[\/latex]\r\n\r\n[latex]6+6\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]\\left(2,0\\right),\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex] and [latex]\\left(4,-3\\right)[\/latex] are all solutions to the equation [latex]3x+2y=6[\/latex]. In the previous example, we found that [latex]\\left(0,3\\right)[\/latex] is a solution, too. We can list these solutions in a table.\r\n<table id=\"fs-id1576667\" class=\"unnumbered\" summary=\"This table it titled 3 x + 2 y = 6. It has 5 rows and 3 columns. The first row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]3x+2y=6[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\Large\\frac{3}{2}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147003[\/ohm_question]\r\n\r\n<\/div>\r\nLet\u2019s find some solutions to another equation now.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind three solutions to the equation [latex]x - 4y=8[\/latex].\r\n[reveal-answer q=\"734894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734894\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468771120\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first starts with the equation x - 4 y = 8. The next line is x = 0, with 0 shown in blue. The next line is 0 - 4 y = 8, with 0 shown in blue. The comment is \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4y=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choose a value for [latex]x[\/latex] or [latex]y[\/latex].<\/td>\r\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\r\n<td>[latex]y=\\color{red}{0}[\/latex]<\/td>\r\n<td>[latex]y=\\color{red}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute it into the equation.<\/td>\r\n<td>[latex]\\color{blue}{0}-4y=8[\/latex]<\/td>\r\n<td>[latex]x-4\\cdot\\color{red}{0}=8[\/latex]<\/td>\r\n<td>[latex]x-4\\cdot\\color{red}{3}=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Solve.<\/td>\r\n<td>[latex]-4y=8[\/latex]\r\n\r\n[latex]y=-2[\/latex]<\/td>\r\n<td>[latex]x-0=8[\/latex]\r\n\r\n[latex]x=8[\/latex]<\/td>\r\n<td>[latex]x-12=8[\/latex]\r\n\r\n[latex]x=20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the ordered pair.<\/td>\r\n<td data-align=\"center\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(20,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]\\left(0,-2\\right),\\left(8,0\\right)[\/latex], and [latex]\\left(20,3\\right)[\/latex] are three solutions to the equation [latex]x - 4y=8[\/latex].\r\n<table id=\"fs-id1580614\" class=\"unnumbered\" summary=\"This table it titled x - 4 y =8. It has 4 rows and 3 columns. The first row is a header row and it labels each column \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]x - 4y=8[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]8[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]20[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(20,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question]147004[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Complete a table of values that satisfy a two variable equation<\/li>\n<li>Find any solution to a two variable equation<\/li>\n<\/ul>\n<\/div>\n<p data-type=\"title\">In the previous examples, we substituted the [latex]x\\text{- and }y\\text{-values}[\/latex] of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[\/latex] and then solve the equation for [latex]y[\/latex]. Or, choose a value for [latex]y[\/latex] and then solve for [latex]x[\/latex].<\/p>\n<p data-type=\"title\">We\u2019ll start by looking at the solutions to the equation [latex]y=5x - 1[\/latex] we found in the previous chapter. We can summarize this information in a table of solutions.<\/p>\n<table id=\"fs-id1801596\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row has the equation y = 5 x -1. The first row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x - 1[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(1,4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\"><\/td>\n<td data-align=\"left\"><\/td>\n<td data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To find a third solution, we\u2019ll let [latex]x=2[\/latex] and solve for [latex]y[\/latex].<\/p>\n<table id=\"eip-id1168469817063\" class=\"unnumbered\" summary=\"The figure shows a substitution into an equation and accompanying comments. The first equation is y = 5 open parentheses 2, shown in blue, closed parentheses - 1. The comment is\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y=5x - 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]x=2[\/latex]<\/td>\n<td>[latex]y=5(\\color{blue}{2})-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]y=10 - 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The ordered pair is a solution to [latex]y=5x - 1[\/latex]. We will add it to the table.<\/p>\n<table id=\"fs-id1569134\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 5 x - 1. The next row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x - 1[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(1,4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(2,9\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can find more solutions to the equation by substituting any value of [latex]x[\/latex] or any value of [latex]y[\/latex] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Complete the table to find three solutions to the equation [latex]y=4x - 2\\text{:}[\/latex]<\/p>\n<table id=\"fs-id1599948\" class=\"unnumbered\" summary=\"This table has 5 rows and 3 columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\"><\/td>\n<td data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\n<td data-align=\"left\"><\/td>\n<td data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\"><\/td>\n<td data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Solution<br \/>\nSubstitute [latex]x=0,x=-1[\/latex], and [latex]x=2[\/latex] into [latex]y=4x - 2[\/latex].<\/p>\n<table id=\"eip-id1168468326216\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation. The first substitution x = 0, with 0 shown in blue. The next line is y = 4 x - 2. The next line is y = 4 times 0, shown in blue, minus 2. The next line is y = 0 - 2. The next line is y = -2. The last line is\" data-label=\"\">\n<tbody>\n<tr>\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-1}[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<td>[latex]y=4x - 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=4\\cdot{\\color{blue}{0}}-2[\/latex]<\/td>\n<td>[latex]y=4(\\color{blue}{-1})-2[\/latex]<\/td>\n<td>[latex]y=4\\cdot{\\color{blue}{2}}-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=0 - 2[\/latex]<\/td>\n<td>[latex]y=-4 - 2[\/latex]<\/td>\n<td>[latex]y=8 - 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]y=-2[\/latex]<\/td>\n<td>[latex]y=-6[\/latex]<\/td>\n<td>[latex]y=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,6\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The results are summarized in the table.<\/p>\n<table id=\"fs-id1572080\" class=\"unnumbered\" summary=\"This table has 5 rows and three columns. The first row is the equation y = 4 x - 2. The next row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]-2[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\n<td data-align=\"left\">[latex]-6[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(-1,-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(2,6\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146945\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146945&theme=oea&iframe_resize_id=ohm146945&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146947\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146947&theme=oea&iframe_resize_id=ohm146947&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Complete the table to find three solutions to the equation [latex]5x - 4y=20\\text{:}[\/latex]<\/p>\n<table id=\"fs-id1328205\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table is 5 rows and 3 columns. The first row is the equation 5 x - 4 y = 20. The next row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px;\" colspan=\"3\" data-align=\"center\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px;\" data-align=\"left\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px;\" data-align=\"left\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\"><\/td>\n<td style=\"width: 114px;\" data-align=\"left\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q471577\">Show Solution<\/span><\/p>\n<div id=\"q471577\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224816\/CNX_BMath_Figure_11_01_038_img.png\" alt=\"The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is\" data-media-type=\"image\/png\" \/><br \/>\nThe results are summarized in the table.<\/p>\n<table id=\"fs-id1572845\" class=\"unnumbered\" style=\"width: 479.75px;\" summary=\"This table has 5 rows and 3 columns. The first row is equation 5 x - 4 y = 20. The next row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px;\" colspan=\"3\" data-align=\"center\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px;\" data-align=\"left\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px;\" data-align=\"left\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px;\" data-align=\"left\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\">[latex]4[\/latex]<\/td>\n<td style=\"width: 114px;\" data-align=\"left\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(4,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px;\" data-align=\"left\">[latex]8[\/latex]<\/td>\n<td style=\"width: 114px;\" data-align=\"left\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px;\" data-align=\"left\">[latex]\\left(8,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146948\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146948&theme=oea&iframe_resize_id=ohm146948&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Find Solutions to Linear Equations in Two Variables<\/h3>\n<p>To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either [latex]x[\/latex] or [latex]y[\/latex]. We could choose [latex]1,100,1,000[\/latex], or any other value we want. But it\u2019s a good idea to choose a number that\u2019s easy to work with. We\u2019ll usually choose [latex]0[\/latex] as one of our values.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find a solution to the equation [latex]3x+2y=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166017\">Show Solution<\/span><\/p>\n<div id=\"q166017\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469770411\" class=\"unnumbered unstyled\" summary=\"The figure shows a four step solution. Step 1 reads\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\"><strong>Step 1:<\/strong> Choose any value for one of the variables in the equation.<\/td>\n<td>We can substitute any value we want for [latex]x[\/latex] or any value for [latex]y[\/latex].<\/p>\n<p>Let&#8217;s pick [latex]x=0[\/latex].<\/p>\n<p>What is the value of [latex]y[\/latex] if [latex]x=0[\/latex] ?<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong>Step 2:<\/strong> Substitute that value into the equation.<\/p>\n<p>Solve for the other variable.<\/td>\n<td>Substitute [latex]0[\/latex] for [latex]x[\/latex].<\/p>\n<p>Simplify.<\/p>\n<p>Divide both sides by [latex]2[\/latex].<\/td>\n<td>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3\\cdot\\color{blue}{0}+2y=6[\/latex]<\/p>\n<p>[latex]0+2y=6[\/latex]<\/p>\n<p>[latex]2y=6[\/latex]<\/p>\n<p>[latex]y=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3:<\/strong> Write the solution as an ordered pair.<\/td>\n<td>So, when [latex]x=0,y=3[\/latex].<\/td>\n<td>This solution is represented by the ordered pair [latex]\\left(0,3\\right)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\"><strong>Step 4:<\/strong> Check.<\/td>\n<td>Substitute [latex]x=\\color{blue}{0}, y=\\color{red}{3}[\/latex] into the equation [latex]3x+2y=6[\/latex]<\/p>\n<p>Is the result a true equation?<\/p>\n<p>Yes!<\/td>\n<td>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3\\cdot\\color{blue}{0}+2\\cdot\\color{red}{3}\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]0+6\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6=6\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147000\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147000&theme=oea&iframe_resize_id=ohm147000&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We said that linear equations in two variables have infinitely many solutions, and we\u2019ve just found one of them. Let\u2019s find some other solutions to the equation [latex]3x+2y=6[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find three more solutions to the equation [latex]3x+2y=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q645203\">Show Solution<\/span><\/p>\n<div id=\"q645203\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo find solutions to [latex]3x+2y=6[\/latex], choose a value for [latex]x[\/latex] or [latex]y[\/latex]. Remember, we can choose any value we want for [latex]x[\/latex] or [latex]y[\/latex]. Here we chose [latex]1[\/latex] for [latex]x[\/latex], and [latex]0[\/latex] and [latex]-3[\/latex] for [latex]y[\/latex].<\/p>\n<table id=\"eip-id1168468473621\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first substitution is y = 0, with 0 shown in red The next line is 3 x + 2 y = 6. The next line is3 x + 2 open parentheses 0, shown in red, closed parentheses = 6. It has the comment\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"bottom\">Substitute it into the equation.<\/td>\n<td>[latex]y=\\color{red}{0}[\/latex]<\/p>\n<p>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3x+2(\\color{red}{0})=6[\/latex]<\/td>\n<td>[latex]y=\\color{blue}{1}[\/latex]<\/p>\n<p>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3(\\color{blue}{1})+2y=6[\/latex]<\/td>\n<td>[latex]y=\\color{red}{-3}[\/latex]<\/p>\n<p>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3x+2(\\color{red}{-3})=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/p>\n<p>Solve.<\/td>\n<td>[latex]3x+0=6[\/latex]<\/p>\n<p>[latex]3x=6[\/latex]<\/td>\n<td>[latex]3+2y=6[\/latex]<\/p>\n<p>[latex]2y=3[\/latex]<\/td>\n<td>[latex]3x-6=6[\/latex]<\/p>\n<p>[latex]3x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]y=\\Large\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]x=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">Write the ordered pair.<\/td>\n<td data-align=\"center\">[latex]\\left(2,0\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Check your answers.<\/p>\n<table id=\"eip-id1168466166098\" class=\"unnumbered unstyled\" summary=\"The figure shows three substitutions into equations. The first starts with\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]\\left(2,0\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3\\cdot\\color{blue}{2}+2\\cdot\\color{red}{0}\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6+0\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6+6\\checkmark[\/latex]<\/td>\n<td>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3\\cdot\\color{blue}{1}+2\\cdot\\color{red}{\\Large\\frac{3}{2}}\\normalsize\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]3+3\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6+6\\checkmark[\/latex]<\/td>\n<td>[latex]3x+2y=6[\/latex]<\/p>\n<p>[latex]3\\cdot\\color{blue}{4}+2\\cdot\\color{red}{-3}\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]12+(-6)\\stackrel{?}{=}6[\/latex]<\/p>\n<p>[latex]6+6\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]\\left(2,0\\right),\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex] and [latex]\\left(4,-3\\right)[\/latex] are all solutions to the equation [latex]3x+2y=6[\/latex]. In the previous example, we found that [latex]\\left(0,3\\right)[\/latex] is a solution, too. We can list these solutions in a table.<\/p>\n<table id=\"fs-id1576667\" class=\"unnumbered\" summary=\"This table it titled 3 x + 2 y = 6. It has 5 rows and 3 columns. The first row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]3x+2y=6[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(2,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\Large\\frac{3}{2}[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]-3[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(4,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147003\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147003&theme=oea&iframe_resize_id=ohm147003&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Let\u2019s find some solutions to another equation now.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find three solutions to the equation [latex]x - 4y=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q734894\">Show Solution<\/span><\/p>\n<div id=\"q734894\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468771120\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions into an equation and accompanying comments. The first starts with the equation x - 4 y = 8. The next line is x = 0, with 0 shown in blue. The next line is 0 - 4 y = 8, with 0 shown in blue. The comment is\" data-label=\"\">\n<tbody>\n<tr>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<td>[latex]x-4y=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Choose a value for [latex]x[\/latex] or [latex]y[\/latex].<\/td>\n<td>[latex]x=\\color{blue}{0}[\/latex]<\/td>\n<td>[latex]y=\\color{red}{0}[\/latex]<\/td>\n<td>[latex]y=\\color{red}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute it into the equation.<\/td>\n<td>[latex]\\color{blue}{0}-4y=8[\/latex]<\/td>\n<td>[latex]x-4\\cdot\\color{red}{0}=8[\/latex]<\/td>\n<td>[latex]x-4\\cdot\\color{red}{3}=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Solve.<\/td>\n<td>[latex]-4y=8[\/latex]<\/p>\n<p>[latex]y=-2[\/latex]<\/td>\n<td>[latex]x-0=8[\/latex]<\/p>\n<p>[latex]x=8[\/latex]<\/td>\n<td>[latex]x-12=8[\/latex]<\/p>\n<p>[latex]x=20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the ordered pair.<\/td>\n<td data-align=\"center\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(8,0\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(20,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]\\left(0,-2\\right),\\left(8,0\\right)[\/latex], and [latex]\\left(20,3\\right)[\/latex] are three solutions to the equation [latex]x - 4y=8[\/latex].<\/p>\n<table id=\"fs-id1580614\" class=\"unnumbered\" summary=\"This table it titled x - 4 y =8. It has 4 rows and 3 columns. The first row is a header row and it labels each column\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]x - 4y=8[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]-2[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]8[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(8,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]20[\/latex]<\/td>\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(20,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147004\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147004&theme=oea&iframe_resize_id=ohm147004&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10674\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID  147004, 147003, 147000. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID  147004, 147003, 147000\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"3bba0eea-8955-4525-9fb6-f8c7cb6d22df","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10674","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10674","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10674\/revisions"}],"predecessor-version":[{"id":15693,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10674\/revisions\/15693"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10674\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=10674"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=10674"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=10674"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=10674"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}