{"id":10673,"date":"2017-06-05T14:58:08","date_gmt":"2017-06-05T14:58:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10673"},"modified":"2020-10-22T09:15:22","modified_gmt":"2020-10-22T09:15:22","slug":"verifying-solutions-to-equations-in-two-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/verifying-solutions-to-equations-in-two-variables\/","title":{"raw":"9.1.c - Finding Solutions to Equations in Two Variables","rendered":"9.1.c &#8211; Finding Solutions to Equations in Two Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether an ordered pair is a solution of an equation<\/li>\r\n \t<li>Complete a table of solutions for a linear equation<\/li>\r\n<\/ul>\r\n<\/div>\r\nAll the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as [latex]x=4[\/latex]. Then we checked the solution by substituting back into the equation.\r\n\r\nHere\u2019s an example of a linear equation in one variable, and its one solution.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}3x+5=17\\hfill \\\\ \\\\ 3x=12\\hfill \\\\ x=4\\hfill \\end{array}[\/latex]<\/p>\r\nBut equations can have more than one variable. Equations with two variables can be written in the general form [latex]Ax+By=C[\/latex]. An equation of this form is called a linear equation in two variables.\r\n<div class=\"textbox shaded\">\r\n<h3>Linear Equation<\/h3>\r\nAn equation of the form [latex]Ax+By=C[\/latex], where [latex]A\\text{ and }B[\/latex] are not both zero, is called a linear equation in two variables.\r\n\r\n<\/div>\r\nNotice that the word \"line\" is in linear.\r\n\r\nHere is an example of a linear equation in two variables, [latex]x[\/latex] and [latex]y\\text{:}[\/latex]\r\n<p style=\"text-align: center\">[latex]\\color{red}{A}x+\\color{blue}{B}y=\\color{green}{C}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]x+\\color{blue}{4}y=\\color{green}{8}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\color{red}{A=1},\\color{blue}{B=4},\\color{green}{C=8}[\/latex]<\/p>\r\nIs [latex]y=-5x+1[\/latex] a linear equation? It does not appear to be in the form [latex]Ax+By=C[\/latex]. But we could rewrite it in this form.\r\n<table id=\"eip-id1168468301870\" class=\"unnumbered unstyled\" summary=\" This image has two columns. The first column on the right is the equation y equals negative 5 times x plus 1. The second line in the left column reads, \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]y=-5x+1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]5x[\/latex] to both sides.<\/td>\r\n<td>[latex]y+5x=-5x+1+5x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y+5x=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to put it in [latex]Ax+By=C[\/latex].<\/td>\r\n<td>[latex]\\color{red}{A}x+\\color{blue}{B}y=C[\/latex]\r\n\r\n[latex]5x+y=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBy rewriting [latex]y=-5x+1[\/latex] as [latex]5x+y=1[\/latex], we can see that it is a linear equation in two variables because it can be written in the form [latex]Ax+By=C[\/latex].\r\n\r\nLinear equations in two variables have infinitely many solutions. For every number that is substituted for [latex]x[\/latex], there is a corresponding [latex]y[\/latex] value. This pair of values is a solution to the linear equation and is represented by the ordered pair [latex]\\left(x,y\\right)[\/latex]. When we substitute these values of [latex]x[\/latex] and [latex]y[\/latex] into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.\r\n<div class=\"textbox shaded\">\r\n<h3>Solution to a Linear Equation in Two Variables<\/h3>\r\nAn ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution to the linear equation [latex]Ax+By=C[\/latex], if the equation is a true statement when the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] of the ordered pair are substituted into the equation.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDetermine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].\r\n\r\n[reveal-answer q=\"980260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"980260\"]For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\r\nEvaluate.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\r\nThe statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].[\/hidden-answer]<span style=\"font-size: 1rem;text-align: initial;background-color: #ffffff\">\u00a0<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine which ordered pairs are solutions of the equation [latex]x+4y=8\\text{:}[\/latex]\r\n\r\n1. [latex]\\left(0,2\\right)[\/latex]\r\n2. [latex]\\left(2,-4\\right)[\/latex]\r\n3. [latex]\\left(-4,3\\right)[\/latex]\r\n\r\nSolution\r\nSubstitute the [latex]x\\text{- and}y\\text{-values}[\/latex] from each ordered pair into the equation and determine if the result is a true statement.\r\n<table id=\"eip-id1168469838906\" class=\"unnumbered unstyled\" summary=\"This image shows three columns. The first column is labeled \">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<td>2. [latex]\\left(2,-4\\right)[\/latex]<\/td>\r\n<td>3. [latex]\\left(-4,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{2}[\/latex]\r\n\r\n[latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{0}+4\\cdot\\color{red}{2}\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]0+8\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]8=8\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{2}, y=\\color{red}{-4}[\/latex]\r\n\r\n[latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{2}+4(\\color{red}{-4})\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]2+(-16)\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]-14\\not=8[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-4}, y=\\color{red}{3}[\/latex]\r\n\r\n[latex]x+4y=8[\/latex]\r\n\r\n[latex]\\color{blue}{-4}+4\\cdot\\color{red}{3}\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]-4+12\\stackrel{?}{=}8[\/latex]\r\n\r\n[latex]8=8\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,2\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(2,-4\\right)[\/latex] is not a solution.<\/td>\r\n<td>[latex]\\left(-4,3\\right)[\/latex] is a solution.<\/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]146928[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine which ordered pairs are solutions of the equation. [latex]y=5x - 1\\text{:}[\/latex]\r\n\r\n1. [latex]\\left(0,-1\\right)[\/latex]\r\n2. [latex]\\left(1,4\\right)[\/latex]\r\n3. [latex]\\left(-2,-7\\right)[\/latex]\r\n[reveal-answer q=\"941200\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"941200\"]\r\n\r\nSolution\r\nSubstitute the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] from each ordered pair into the equation and determine if it results in a true statement.\r\n<table id=\"eip-id1168466112848\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions in three columns, a, b, and c. The equation is y equals 5 times x minus 1. The first substitution is for ordered pair (0, -1). The first line in column a reads x = 0, with 0 shown in blue and y = -1, with -1 shown in red. The next line is y = 5 x - 1. The next line is -1, shown in red = 5 open parentheses 0, shown in blue, closed parentheses - 1, with a question mark shown over the equal sign. The next line is -1 = 0 - 1, with a question mark over the equal sign. The next line is - 1 = -1, followed by a check mark. The last line is \">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<td>2. [latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<td>3. [latex]\\left(-2,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{-1}[\/latex]\r\n\r\n[latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{-1}\\stackrel{?}{=}5(\\color{blue}{0})-1[\/latex]\r\n\r\n[latex]-1\\stackrel{?}{=}0-1[\/latex]\r\n\r\n[latex]-1=-1\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{1}, y=\\color{red}{4}[\/latex]\r\n\r\n[latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{4}\\stackrel{?}{=}5(\\color{blue}{1})-1[\/latex]\r\n\r\n[latex]4\\stackrel{?}{=}5-1[\/latex]\r\n\r\n[latex]4=4\\checkmark[\/latex]<\/td>\r\n<td>[latex]x=\\color{blue}{-2}, y=\\color{red}{-7}[\/latex]\r\n\r\n[latex]y=5x-1[\/latex]\r\n\r\n[latex]\\color{red}{-7}\\stackrel{?}{=}5(\\color{blue}{-2})-1[\/latex]\r\n\r\n[latex]-7\\stackrel{?}{=}-10-1[\/latex]\r\n\r\n[latex]-7\\not=-11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(0,-1\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(1,4\\right)[\/latex] is a solution.<\/td>\r\n<td>[latex]\\left(-2,-7\\right)[\/latex] is not a solution.<\/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]146929[\/ohm_question]\r\n\r\n[ohm_question]146941[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video you will see more\u00a0examples of how to determine whether an ordered pair is a solution to a linear equation.\r\n\r\nhttps:\/\/youtu.be\/9aWGxt7OnB8\r\n<h2>Complete a Table of Solutions to a Linear Equation<\/h2>\r\nIn 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].\r\n\r\nWe\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.\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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/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 \">\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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]\\left(1,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/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 \">\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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-6[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px\">[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\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px\"><\/td>\r\n<td style=\"width: 218.75px\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px\"><\/td>\r\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px\"><\/td>\r\n<td style=\"width: 114px\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px\"><\/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 \" \/>\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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 446.75px\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th style=\"width: 114px\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 114px\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 218.75px\">[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\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 114px\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 218.75px\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 218.75px\">[latex]\\left(4,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 114px\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 114px\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 218.75px\">[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 \">\r\n<tbody>\r\n<tr>\r\n<td><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><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><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 \">\r\n<tbody>\r\n<tr>\r\n<td>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>Write the ordered pair.<\/td>\r\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<td>[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 \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<td>[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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]3x+2y=6[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]\\left(0,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\\Large\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[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 \">\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>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<td>[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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]x - 4y=8[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[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>Determine whether an ordered pair is a solution of an equation<\/li>\n<li>Complete a table of solutions for a linear equation<\/li>\n<\/ul>\n<\/div>\n<p>All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as [latex]x=4[\/latex]. Then we checked the solution by substituting back into the equation.<\/p>\n<p>Here\u2019s an example of a linear equation in one variable, and its one solution.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}3x+5=17\\hfill \\\\ \\\\ 3x=12\\hfill \\\\ x=4\\hfill \\end{array}[\/latex]<\/p>\n<p>But equations can have more than one variable. Equations with two variables can be written in the general form [latex]Ax+By=C[\/latex]. An equation of this form is called a linear equation in two variables.<\/p>\n<div class=\"textbox shaded\">\n<h3>Linear Equation<\/h3>\n<p>An equation of the form [latex]Ax+By=C[\/latex], where [latex]A\\text{ and }B[\/latex] are not both zero, is called a linear equation in two variables.<\/p>\n<\/div>\n<p>Notice that the word &#8220;line&#8221; is in linear.<\/p>\n<p>Here is an example of a linear equation in two variables, [latex]x[\/latex] and [latex]y\\text{:}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\color{red}{A}x+\\color{blue}{B}y=\\color{green}{C}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]x+\\color{blue}{4}y=\\color{green}{8}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\color{red}{A=1},\\color{blue}{B=4},\\color{green}{C=8}[\/latex]<\/p>\n<p>Is [latex]y=-5x+1[\/latex] a linear equation? It does not appear to be in the form [latex]Ax+By=C[\/latex]. But we could rewrite it in this form.<\/p>\n<table id=\"eip-id1168468301870\" class=\"unnumbered unstyled\" summary=\"This image has two columns. The first column on the right is the equation y equals negative 5 times x plus 1. The second line in the left column reads,\">\n<tbody>\n<tr>\n<td>[latex]y=-5x+1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]5x[\/latex] to both sides.<\/td>\n<td>[latex]y+5x=-5x+1+5x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y+5x=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to put it in [latex]Ax+By=C[\/latex].<\/td>\n<td>[latex]\\color{red}{A}x+\\color{blue}{B}y=C[\/latex]<\/p>\n<p>[latex]5x+y=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>By rewriting [latex]y=-5x+1[\/latex] as [latex]5x+y=1[\/latex], we can see that it is a linear equation in two variables because it can be written in the form [latex]Ax+By=C[\/latex].<\/p>\n<p>Linear equations in two variables have infinitely many solutions. For every number that is substituted for [latex]x[\/latex], there is a corresponding [latex]y[\/latex] value. This pair of values is a solution to the linear equation and is represented by the ordered pair [latex]\\left(x,y\\right)[\/latex]. When we substitute these values of [latex]x[\/latex] and [latex]y[\/latex] into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.<\/p>\n<div class=\"textbox shaded\">\n<h3>Solution to a Linear Equation in Two Variables<\/h3>\n<p>An ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution to the linear equation [latex]Ax+By=C[\/latex], if the equation is a true statement when the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] of the ordered pair are substituted into the equation.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Determine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q980260\">Show Solution<\/span><\/p>\n<div id=\"q980260\" class=\"hidden-answer\" style=\"display: none\">For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\n<p>Evaluate.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\n<p>The statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p><\/div>\n<\/div>\n<p><span style=\"font-size: 1rem;text-align: initial;background-color: #ffffff\">\u00a0<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine which ordered pairs are solutions of the equation [latex]x+4y=8\\text{:}[\/latex]<\/p>\n<p>1. [latex]\\left(0,2\\right)[\/latex]<br \/>\n2. [latex]\\left(2,-4\\right)[\/latex]<br \/>\n3. [latex]\\left(-4,3\\right)[\/latex]<\/p>\n<p>Solution<br \/>\nSubstitute the [latex]x\\text{- and}y\\text{-values}[\/latex] from each ordered pair into the equation and determine if the result is a true statement.<\/p>\n<table id=\"eip-id1168469838906\" class=\"unnumbered unstyled\" summary=\"This image shows three columns. The first column is labeled\">\n<tbody>\n<tr>\n<td>1. [latex]\\left(0,2\\right)[\/latex]<\/td>\n<td>2. [latex]\\left(2,-4\\right)[\/latex]<\/td>\n<td>3. [latex]\\left(-4,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{2}[\/latex]<\/p>\n<p>[latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{0}+4\\cdot\\color{red}{2}\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]0+8\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]8=8\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{2}, y=\\color{red}{-4}[\/latex]<\/p>\n<p>[latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{2}+4(\\color{red}{-4})\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]2+(-16)\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]-14\\not=8[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-4}, y=\\color{red}{3}[\/latex]<\/p>\n<p>[latex]x+4y=8[\/latex]<\/p>\n<p>[latex]\\color{blue}{-4}+4\\cdot\\color{red}{3}\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]-4+12\\stackrel{?}{=}8[\/latex]<\/p>\n<p>[latex]8=8\\checkmark[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,2\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(2,-4\\right)[\/latex] is not a solution.<\/td>\n<td>[latex]\\left(-4,3\\right)[\/latex] is a solution.<\/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=\"ohm146928\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146928&theme=oea&iframe_resize_id=ohm146928&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>Determine which ordered pairs are solutions of the equation. [latex]y=5x - 1\\text{:}[\/latex]<\/p>\n<p>1. [latex]\\left(0,-1\\right)[\/latex]<br \/>\n2. [latex]\\left(1,4\\right)[\/latex]<br \/>\n3. [latex]\\left(-2,-7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q941200\">Show Solution<\/span><\/p>\n<div id=\"q941200\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nSubstitute the [latex]x\\text{-}[\/latex] and [latex]y\\text{-values}[\/latex] from each ordered pair into the equation and determine if it results in a true statement.<\/p>\n<table id=\"eip-id1168466112848\" class=\"unnumbered unstyled\" summary=\"The figure shows three algebraic substitutions in three columns, a, b, and c. The equation is y equals 5 times x minus 1. The first substitution is for ordered pair (0, -1). The first line in column a reads x = 0, with 0 shown in blue and y = -1, with -1 shown in red. The next line is y = 5 x - 1. The next line is -1, shown in red = 5 open parentheses 0, shown in blue, closed parentheses - 1, with a question mark shown over the equal sign. The next line is -1 = 0 - 1, with a question mark over the equal sign. The next line is - 1 = -1, followed by a check mark. The last line is\">\n<tbody>\n<tr>\n<td>1. [latex]\\left(0,-1\\right)[\/latex]<\/td>\n<td>2. [latex]\\left(1,4\\right)[\/latex]<\/td>\n<td>3. [latex]\\left(-2,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x=\\color{blue}{0}, y=\\color{red}{-1}[\/latex]<\/p>\n<p>[latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{-1}\\stackrel{?}{=}5(\\color{blue}{0})-1[\/latex]<\/p>\n<p>[latex]-1\\stackrel{?}{=}0-1[\/latex]<\/p>\n<p>[latex]-1=-1\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{1}, y=\\color{red}{4}[\/latex]<\/p>\n<p>[latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{4}\\stackrel{?}{=}5(\\color{blue}{1})-1[\/latex]<\/p>\n<p>[latex]4\\stackrel{?}{=}5-1[\/latex]<\/p>\n<p>[latex]4=4\\checkmark[\/latex]<\/td>\n<td>[latex]x=\\color{blue}{-2}, y=\\color{red}{-7}[\/latex]<\/p>\n<p>[latex]y=5x-1[\/latex]<\/p>\n<p>[latex]\\color{red}{-7}\\stackrel{?}{=}5(\\color{blue}{-2})-1[\/latex]<\/p>\n<p>[latex]-7\\stackrel{?}{=}-10-1[\/latex]<\/p>\n<p>[latex]-7\\not=-11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(0,-1\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(1,4\\right)[\/latex] is a solution.<\/td>\n<td>[latex]\\left(-2,-7\\right)[\/latex] is not a solution.<\/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=\"ohm146929\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146929&theme=oea&iframe_resize_id=ohm146929&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146941\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146941&theme=oea&iframe_resize_id=ohm146941&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video you will see more\u00a0examples of how to determine whether an ordered pair is a solution to a linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine If an Ordered Pair is a Solution to a Linear Equation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9aWGxt7OnB8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Complete a Table of Solutions to a Linear Equation<\/h2>\n<p>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>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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\left(1,4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td><\/td>\n<td><\/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\">\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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=5x - 1[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\left(1,4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]-1[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td><\/td>\n<td><\/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\">\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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]y=4x - 2[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-6[\/latex]<\/td>\n<td>[latex]\\left(-1,-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[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\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px\"><\/td>\n<td style=\"width: 218.75px\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px\"><\/td>\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px\"><\/td>\n<td style=\"width: 114px\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px\"><\/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\" \/><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\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 446.75px\" colspan=\"3\">[latex]5x - 4y=20[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 114px\">[latex]x[\/latex]<\/th>\n<th style=\"width: 114px\">[latex]y[\/latex]<\/th>\n<th style=\"width: 218.75px\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 114px\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 218.75px\">[latex]\\left(0,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px\">[latex]4[\/latex]<\/td>\n<td style=\"width: 114px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 218.75px\">[latex]\\left(4,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 114px\">[latex]8[\/latex]<\/td>\n<td style=\"width: 114px\">[latex]5[\/latex]<\/td>\n<td style=\"width: 218.75px\">[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\">\n<tbody>\n<tr>\n<td><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><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><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\">\n<tbody>\n<tr>\n<td>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>Write the ordered pair.<\/td>\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<td>[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\">\n<tbody>\n<tr>\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<td>[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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]3x+2y=6[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]\\left(0,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(2,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\Large\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\Large\\frac{3}{2}\\normalsize\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[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\">\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>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\n<td>[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\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]x - 4y=8[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\left(0,-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(8,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[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-10673\">\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 146941, 146929, 146928, 146927. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Determine If An Ordered Pair is a Solution to a Linear Equation. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pJtxugdFjEk\">https:\/\/youtu.be\/pJtxugdFjEk<\/a>. <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":5,"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 146941, 146929, 146928, 146927\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine If An Ordered Pair is a Solution to a Linear Equation\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pJtxugdFjEk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"04dcacf26ac2496eb29551e7708e024a, 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