{"id":18213,"date":"2022-04-07T19:11:14","date_gmt":"2022-04-07T19:11:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18213"},"modified":"2022-04-28T23:14:42","modified_gmt":"2022-04-28T23:14:42","slug":"cr-4-solving-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-4-solving-linear-equations\/","title":{"raw":"CR.4: Solving Linear Equations","rendered":"CR.4: Solving Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether a number is a solution to an equation<\/li>\r\n \t<li>Review the use of the subtraction and addition properties of equality to solve linear equations<\/li>\r\n \t<li>Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality<\/li>\r\n \t<li>Check your solution to a linear equation to verify its accuracy<\/li>\r\n \t<li>Review and use the division and multiplication properties of equality to solve linear equations<\/li>\r\n \t<li>Use a reciprocal to solve a linear equation that contains fractions<\/li>\r\n \t<li>Solve a linear equation that requires simplification before using properties of equality<\/li>\r\n \t<li>Solve a linear equation that requires a combination of the properties of equality<\/li>\r\n \t<li>Solve linear equations by isolating constants and variables<\/li>\r\n \t<li>Solve linear equations with variables on both sides that require several steps<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Using the Subtraction and Addition Properties for Single-Step Equations<\/h2>\r\nWe began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">Solution of an Equation<\/h3>\r\nA solution of an equation is a value of a variable that makes a true statement when substituted into the equation.\r\n\r\nIn the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.\r\n\r\nDetermine whether a number is a solution to an equation.\r\n<ol id=\"eip-95\" class=\"stepwise\">\r\n \t<li>Substitute the number for the variable in the equation.<\/li>\r\n \t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n \t<li>Determine whether the resulting equation is true.\r\n<ul id=\"fs-id1703984\">\r\n \t<li>If it is true, the number is a solution.<\/li>\r\n \t<li>If it is not true, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nDetermine whether [latex]y=\\frac{3}{4}[\/latex] is a solution for [latex]4y+3=8y[\/latex].\r\n\r\nSolution:\r\n<table id=\"eip-id1168466426761\" class=\"unnumbered unstyled\" summary=\"The top line says 4y plus 3 equals 8y. Beside this is \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4y+3=8y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{\\frac{3}{4}}[\/latex] for [latex]y[\/latex]<\/td>\r\n<td>[latex]4(\\color{red}{\\frac{3}{4}})+3\\stackrel{\\text{?}}{=}8(\\color{red}{\\frac{3}{4}})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+3\\stackrel{\\text{?}}{=}6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]6=6\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]y=\\frac{3}{4}[\/latex] results in a true equation, [latex]\\frac{3}{4}[\/latex] is a solution to the equation [latex]4y+3=8y[\/latex].\r\n\r\n<\/div>\r\nNow it is your turn to determine whether a fraction is the solution to an equation.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141717&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"380\"><\/iframe>\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141719&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"380\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1956928\"><strong>Subtraction Property of Equality<\/strong><\/h3>\r\nFor all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1166490863495\"><strong>Addition Property of Equality<\/strong><\/h3>\r\nFor all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].\r\n\r\n<\/div>\r\nThe goal is to isolate the variable on one side of the equation.\r\n\r\nSome people picture a balance scale, as in the image below, when they solve equations.\r\n<figure id=\"CNX_BMath_Figure_08_01_001\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222535\/CNX_BMath_Figure_08_01_001.png\" alt=\"Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is \" \/><\/figure>\r\nThe quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.\r\n\r\n&nbsp;\r\n\r\nIn the following example we\u00a0review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation to make sure you have a true statement.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]x+11=-3[\/latex].\r\n[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190834\"]\r\n\r\nSolution:\r\nTo isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.\r\n<table id=\"eip-id1168468291100\" class=\"unnumbered unstyled\" summary=\"The top line says x plus 11 equals negative 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]x+11=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Subtract 11 from each side to \"undo\" the addition.<\/td>\r\n<td>[latex]x+11\\color{red}{-11}=-3\\color{red}{-11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]x=-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]x+11=-3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]x=-14[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{-14}+11\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try solving an equation that requires using the addition property.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n<iframe id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141721&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the original equation in the previous example, [latex]11[\/latex] was added to the [latex]x[\/latex] , so we subtracted [latex]11[\/latex] to \"undo\" the addition. In the next example, we will need to \"undo\" subtraction by using the Addition Property of Equality.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]m - 4=-5[\/latex].\r\n[reveal-answer q=\"604060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604060\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467157298\" class=\"unnumbered unstyled\" summary=\"The first line says m minus 4 equals negative 5. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]m-4=-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add 4 to each side to \"undo\" the subtraction.<\/td>\r\n<td>[latex]m-4\\color{red}{+4}=-5\\color{red}{+4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]m=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]m-4=-5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]m=-1[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{-1}+4\\stackrel{\\text{?}}{=}-5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5=-5\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>The solution to [latex]m - 4=-5[\/latex] is [latex]m=-1[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try using the addition property to solve an equation.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom27\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141723&amp;theme=oea&amp;iframe_resize_id=mom27\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video, we present more examples of solving equations using the addition and subtraction properties.\r\n\r\nhttps:\/\/youtu.be\/yqdlj0lv7Cc\r\n<h2>Using the Subtraction and Addition Properties for Multi-Step Equations<\/h2>\r\nIn the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve:\r\n\r\n[latex]3x - 7 - 2x - 4=1[\/latex].\r\n\r\nSolution:\r\nThe left side of the equation has an expression that we should simplify before trying to isolate the variable.\r\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\r\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]x-11=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\r\n<td>[latex]x-11\\color{red}{+11}=1\\color{red}{+11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.\r\n\r\nSubstitute [latex]x=12[\/latex] into the original equation.\r\n[latex]3x-7-2x-4=1[\/latex]\r\n\r\n[latex]3(\\color{red}{12})-7-2(\\color{red}{12})-4=1[\/latex]\r\n\r\n[latex]36-7-24-4=1[\/latex]\r\n\r\n[latex]29-24-4=1[\/latex]\r\n\r\n[latex]5-4=1[\/latex]\r\n\r\n[latex]1=1\\quad\\checkmark[\/latex]\r\n\r\nThe solution checks.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow you can try solving a couple\u00a0equations where you should simplify first.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe last few examples involved simplifying using addition and subtraction. Let's look at an example where we need to distribute first in order to simplify the equation as much as possible.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]3\\left(n - 4\\right)-2n=-3[\/latex].\r\n[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190834\"]\r\n\r\nSolution:\r\nThe left side of the equation has an expression that we should simplify.\r\n<table id=\"eip-id1168468254328\" class=\"unnumbered unstyled\" summary=\"The top line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3(n-4)-2n=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute on the left.<\/td>\r\n<td>[latex]3n-12-2n=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to rearrange terms.<\/td>\r\n<td>[latex]3n-2n-12=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]n-12=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Isolate <em>n<\/em> using the Addition Property of Equality.<\/td>\r\n<td>[latex]n-12\\color{red}{+12}=-3\\color{red}{+12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]n=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.\r\n\r\nSubstitute [latex]n=9[\/latex] into the original equation.\r\n[latex]3(n-4)-2n=-3[\/latex]\r\n[latex]3(\\color{red}{9}-4)-2\\cdot\\color{red}{9}=-3[\/latex]\r\n[latex]3(5)-18=-3[\/latex]\r\n[latex]15-18=-3[\/latex]\r\n[latex]-3=-3\\quad\\checkmark[\/latex]\r\nThe solution checks.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a few problems that involve distribution.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141737&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe next example has expressions on both sides that need to be simplified.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex].\r\n<p class=\"p1\">[reveal-answer q=\"190976\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190976\"]<\/p>\r\nSolution:\r\nBoth sides of the equation have expressions that we should simplify before we isolate the variable.\r\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute on the left, subtract on the right.<\/td>\r\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property of Addition.<\/td>\r\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\r\n<td>[latex]k-2\\color{red}{+2}=-9\\color{red}{+2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]k=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.\r\n\r\nLet [latex]k=-7[\/latex].\r\n[latex]2(3k-1)-5k=-2-7[\/latex]\r\n[latex]2(3(\\color{red}{-7}-1)-5(\\color{red}{-7})=-2-7[\/latex]\r\n[latex]2(-21-1)-5(-7)=-9[\/latex]\r\n[latex]2(-22)+35=-9[\/latex]\r\n[latex]-44+35=-9[\/latex]\r\n[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe solution checks.\r\n<p class=\"p1\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nNow, you give it a try!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom220\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141739&amp;theme=oea&amp;iframe_resize_id=mom220\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the following video we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.\r\n\r\nhttps:\/\/youtu.be\/shGKzDBA5kQ\r\n<h2>Using the Division and Multiplication Properties of Equality for Single-Step Equations<\/h2>\r\nLet's review the Division and Multiplication Properties of Equality as we prepare to use them to solve single-step equations.\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">Division Property of Equality<\/h3>\r\nFor all real numbers [latex]a,b,c[\/latex], and [latex]c\\ne 0[\/latex], if [latex]a=b[\/latex], then [latex]\\frac{a}{c}=\\frac{b}{c}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">Multiplication Property of Equality<\/h3>\r\nFor all real numbers [latex]a,b,c[\/latex], if [latex]a=b[\/latex], then [latex]ac=bc[\/latex].\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nStated simply, when you divide or multiply both sides of an equation by the same quantity, you still have equality.\r\n\r\nLet\u2019s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to \"undo\" the operation on the variable. In the example below the variable is multiplied by [latex]4[\/latex], so we will divide both sides by [latex]4[\/latex] to \"undo\" the multiplication.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]4x=-28[\/latex].\r\n\r\nSolution:\r\n\r\nTo solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4x=-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by 4 to undo the multiplication.<\/td>\r\n<td>[latex]\\frac{4x}{\\color{red}4}=\\frac{-28}{\\color{red}4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x =-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer.<\/td>\r\n<td>[latex]4x=-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=-7[\/latex]. Substitute [latex]-7[\/latex] for x.<\/td>\r\n<td>[latex]4(\\color{red}{-7})\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]-28=-28[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince this is a true statement, [latex]x=-7[\/latex] is a solution to [latex]4x=-28[\/latex].\r\n\r\n<\/div>\r\nNow you can try to solve an equation that requires division and\u00a0includes negative numbers.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141857&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the previous example, to \"undo\" multiplication, we divided. How do you think we \"undo\" division? Next, we will show an example that requires us to use multiplication to undo division.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]\\frac{a}{-7}=-42[\/latex].\r\n\r\n[reveal-answer q=\"399032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399032\"]\r\n\r\nSolution:\r\nHere [latex]a[\/latex] is divided by [latex]-7[\/latex]. We can multiply both sides by [latex]-7[\/latex] to isolate [latex]a[\/latex].\r\n<table id=\"eip-id1168468288515\" class=\"unnumbered unstyled\" summary=\"The top shows a over negative 7 equals negative 42. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\frac{a}{-7}=-42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply both sides by [latex]-7[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{-7}(\\frac{a}{-7})=\\color{red}{-7}(-42)[\/latex]\r\n\r\n[latex]\\frac{-7a}{-7}=294[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]a=294[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer.<\/td>\r\n<td>[latex]\\frac{a}{-7}=-42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]a=294[\/latex] .<\/td>\r\n<td>[latex]\\frac{\\color{red}{294}}{-7}\\stackrel{\\text{?}}{=}-42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-42=-42\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow see if you can solve a\u00a0problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers - two negatives give a positive when they are multiplied.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom21\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141868&amp;theme=oea&amp;iframe_resize_id=mom21\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nAs you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable. \u00a0As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]-r=2[\/latex].\r\n\r\n[reveal-answer q=\"388033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388033\"]\r\n\r\nSolution:\r\nRemember [latex]-r[\/latex] is equivalent to [latex]-1r[\/latex].\r\n<table id=\"eip-id1168469604717\" class=\"unnumbered unstyled\" summary=\"The first line says negative r equals 2. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]-r=2[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite [latex]-r[\/latex] as [latex]-1r[\/latex] .<\/td>\r\n<td>[latex]-1r=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by [latex]-1[\/latex] .<\/td>\r\n<td>[latex]\\frac{-1r}{\\color{red}{-1}}=\\frac{2}{\\color{red}{-1}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]r=-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.<\/td>\r\n<td>[latex]-r=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]r=-2[\/latex]<\/td>\r\n<td>[latex]-(\\color{red}{-2})\\stackrel{\\text{?}}{=}2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]2=2\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try to solve an equation with a negative variable.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141865&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nIn our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]\\frac{2}{3}x=18[\/latex].\r\n\r\n[reveal-answer q=\"444022\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"444022\"]\r\n\r\nSolution:\r\nSince the product of a number and its reciprocal is [latex]1[\/latex], our strategy will be to isolate [latex]x[\/latex] by multiplying by the reciprocal of [latex]\\frac{2}{3}[\/latex].\r\n<table id=\"eip-id1168468646645\" class=\"unnumbered unstyled\" summary=\"The first line shows two-thirds x equals 18. The next line says \">\r\n<tbody>\r\n<tr style=\"height: 37px;\">\r\n<td style=\"height: 37px;\">[latex]\\frac{2}{3}x=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 38px;\">\r\n<td style=\"height: 38px;\">Multiply by the reciprocal of [latex]\\frac{2}{3}[\/latex] .<\/td>\r\n<td style=\"height: 38px;\">[latex]\\frac{\\color{red}{3}}{\\color{red}{2}}\\cdot\\frac{2}{3}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 37px;\">\r\n<td style=\"height: 37px;\">Reciprocals multiply to one.<\/td>\r\n<td style=\"height: 37px;\">[latex]1x=\\frac{3}{2}\\cdot\\frac{18}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px;\">Multiply.<\/td>\r\n<td style=\"height: 18px;\">[latex]x=27[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Check your answer.<\/td>\r\n<td style=\"height: 44.8594px;\">[latex]\\frac{2}{3}x=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44.8594px;\">\r\n<td style=\"height: 14px;\">Let [latex]x=27[\/latex].<\/td>\r\n<td style=\"height: 45px;\">[latex]\\frac{2}{3}\\cdot\\color{red}{27}\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td><\/td>\r\n<td style=\"height: 24px;\">[latex]18=18\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that we could have divided both sides of the equation [latex]\\frac{2}{3}x=18[\/latex] by [latex]\\frac{2}{3}[\/latex] to isolate [latex]x[\/latex]. While this would work, multiplying by the reciprocal requires fewer steps.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141871&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"260\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.\r\n\r\nhttps:\/\/youtu.be\/TB1rkPbF8rA\r\n<h2>Using the Division and Multiplication Properties of Equality for Multi-Step Equations<\/h2>\r\nMany equations start out more complicated than the ones we\u2019ve just solved. Our goal has been to familiarize you with the many ways to apply the addition, subtraction, multiplication, and division properties that are used to solve equations algebraically. Let's work through an example that will employ the following techniques:\r\n<ul>\r\n \t<li>simplify by combining like terms<\/li>\r\n \t<li>isolate x by using the division property of equality<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve: [latex]8x+9x - 5x=-3+15[\/latex].\r\n\r\nSolution:\r\n\r\nFirst, we need to simplify both sides of the equation as much as possible\r\n\r\nStart by combining like terms to simplify each side.\r\n<table id=\"eip-id1168466098204\" class=\"unnumbered unstyled\" summary=\"The first line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8x+9x-5x=-3+15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]12x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by 12 to isolate x.<\/td>\r\n<td>[latex]\\frac{12x}{\\color{red}{12}}=\\frac{12}{\\color{red}{12}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer. Let [latex]x=1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8x+9x-5x=-3+15[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8\\cdot\\color{red}{1}+9\\cdot\\color{red}{1}-5\\cdot\\color{red}{1}\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]8+9-5\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]12=12\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nHere is a similar problem for you to try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try\u00a0it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141884&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nYou may not always have the variables on the left side of the equation, so we will show an example with variables on the right side. You will see that the properties used to solve this equation are exactly the same as the previous example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]11 - 20=17y - 8y - 6y[\/latex].\r\n\r\n[reveal-answer q=\"399032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399032\"]\r\n\r\nSolution:\r\n\r\nSimplify each side by combining like terms.\r\n<table id=\"eip-id1168466111452\" class=\"unnumbered unstyled\" summary=\"The first line shows 11 minus 20 equals 17y minus 8y minus 6y. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11-20=17y-8y-6y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each side.<\/td>\r\n<td>[latex]-9=3y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by 3 to isolate y.<\/td>\r\n<td>[latex]\\frac{-9}{\\color{red}{3}}=\\frac{3y}{\\color{red}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-3=y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer. Let [latex]y=-3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11-20=17y-8y-6y[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11-20\\stackrel{\\text{?}}{=}17(\r\n\\color{red}{-3})-8(\\color{red}{-3})-6(\\color{red}{-3})[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11-20\\stackrel{\\text{?}}{=}-51+24+18[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.\r\n\r\nNow you can try solving a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom23\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141901&amp;theme=oea&amp;iframe_resize_id=mom23\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nIn our next example, we have an equation that contains a set of parentheses. \u00a0We will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]-3\\left(n - 2\\right)-6=21[\/latex].\r\n\r\nRemember\u2014always simplify each side first.\r\n[reveal-answer q=\"789987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"789987\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468278405\" class=\"unnumbered unstyled\" summary=\"The first line shows negative 3 times parentheses n minus 2 minus 6 equals 21. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]-3n+6-6=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-3n=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by -3 to isolate n.<\/td>\r\n<td>[latex]\\frac{-3n}{\\color{red}{-3}}=\\frac{21}{\\color{red}{-3}}[\/latex]\r\n\r\n[latex]n=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer. Let [latex]n=-7[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(\\color{red}{-7}-2)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(-9)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]27-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom27\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141911&amp;theme=oea&amp;iframe_resize_id=mom27\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following\u00a0video you will see another example of using the division property of equality to solve an equation as well as \u00a0another example of how to solve a multi-step equation that includes a set of parentheses.\r\n\r\nhttps:\/\/youtu.be\/qe89pkRKzRw\r\n<h2>Solving Equations With Variables on Both Sides<\/h2>\r\nThe equations we solved in the last section\u00a0simplified nicely so that we could use the division property to isolate the variable and solve the equation. Sometimes, after you simplify you may have a variable and a constant term on the same side of the equal sign.\r\n\r\nOur strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate\u00a0the variable terms on one side of the equation.\r\n\r\nRead on to find out how to solve this kind of equation.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nSolve: [latex]4x+6=-14[\/latex].\r\n\r\nSolution:\r\n\r\nIn this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.\r\n<table style=\"width: 70%;\" summary=\"The top line says 4x plus 6 equals negative 14.\">\r\n<tbody>\r\n<tr style=\"height: 45.8594px;\">\r\n<td style=\"height: 45.8594px; width: 1179.02px;\" colspan=\"2\">Since the left side is the variable side, the 6 is out of place. We must \"undo\" adding [latex]6[\/latex] by subtracting [latex]6[\/latex], and to keep the equality we must subtract [latex]6[\/latex] from both sides. Use the Subtraction Property of Equality.<\/td>\r\n<td style=\"height: 45.8594px; width: 176px;\">[latex]4x+6\\color{red}{-6}=-14\\color{red}{-6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]4x=-20[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Now all the [latex]x[\/latex] s are on the left and the constant on the right.<\/td>\r\n<td style=\"height: 15px; width: 176px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 60px;\">\r\n<td style=\"height: 60px; width: 1179.02px;\" colspan=\"2\">Use the Division Property of Equality.<\/td>\r\n<td style=\"height: 60px; width: 176px;\">[latex]\\frac{4x}{\\color{red}{4}}=\\frac{-20}{\\color{red}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]x=-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Check:<\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]4x+6=-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Let [latex]x=-5[\/latex] .<\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]4(\\color{red}{-5})+6=-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]-20+6=-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15px; width: 176px;\">[latex]-14=-14\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSolve: [latex]2y - 7=15[\/latex].\r\n[reveal-answer q=\"629971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"629971\"]\r\n\r\nSolution:\r\nNotice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the [latex]7[\/latex] is out of place. It is subtracted from the [latex]2y[\/latex], so to \"undo\" subtraction, add [latex]7[\/latex] to both sides.\r\n<table id=\"eip-id1168469592645\" class=\"unnumbered unstyled\" summary=\"The first line says 2y minus 7 equals 15. The left side is labeled \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]2y-7[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span>\r\n\r\n<span style=\"color: #000000;\">[latex]15[\/latex] is the side containing only a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add [latex]7[\/latex] to both sides.<\/td>\r\n<td>[latex]2y-7\\color{red}{+7}=15\\color{red}{+7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]2y=22[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"3\">The variables are now on one side and the constants on the other.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Divide both sides by [latex]2[\/latex].<\/td>\r\n<td>[latex]\\frac{2y}{\\color{red}{2}}=\\frac{22}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]y=11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Check:<\/td>\r\n<td>[latex]2y-7=15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Let [latex]y=11[\/latex] .<\/td>\r\n<td>[latex]2\\cdot\\color{red}{11}-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]22-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]15=15\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try\u00a0a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try\u00a0It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142131&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Solve Equations with Variables on Both Sides<\/h3>\r\nYou may have noticed that in all the equations we have solved so far, we had\u00a0variables on only one\u00a0side of the equation. This does not happen all the time\u2014so now we\u2019ll see how to solve equations where there are variable terms on both sides of the equation. We will start like we did above\u2014choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side as well.\r\n\r\nIn the next example,\u00a0the variable, [latex]x[\/latex], is on both sides, but the constants appear only on the right side, so we'll make the right side the \"constant\" side. Then the left side will be the \"variable\" side.\r\n<div class=\"textbox exercises\">\r\n<h3>ExampleS<\/h3>\r\nSolve: [latex]5x=4x+7[\/latex].\r\n[reveal-answer q=\"235739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"235739\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168465095460\" class=\"unnumbered unstyled\" summary=\"The first line says 5x equals 4x plus 7. The left side is labeled \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">[latex]5x[\/latex] is the side containing only a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><span style=\"color: #000000;\">[latex]4x+7[\/latex] is the side containing a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">We don't want any variables on the right, so subtract the [latex]4x[\/latex] .<\/td>\r\n<td>[latex]5x\\color{red}{-4x}=4x\\color{red}{-4x}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]x=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">We have all the variables on one side and the constants on the other. We have solved the equation.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td><\/td>\r\n<td>[latex]5x=4x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]7[\/latex] for [latex]x[\/latex] .<\/td>\r\n<td><\/td>\r\n<td>[latex]5(\\color{red}{7})\\stackrel{\\text{?}}{=}4(\\color{red}{7})+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]35\\stackrel{\\text{?}}{=}28+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex]35=35\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSolve: [latex]7x=-x+24[\/latex].\r\n[reveal-answer q=\"192799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"192799\"]\r\n\r\nSolution:\r\nThe only constant, [latex]24[\/latex], is on the right, so let the left side be the variable side.\r\n<table id=\"eip-id1168467377760\" class=\"unnumbered unstyled\" summary=\"The first line says 7x equals negative x plus 24. The left side is labeled \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]7x[\/latex] is the side containing only a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span>\r\n\r\n<span style=\"color: #000000;\">[latex]-x+24[\/latex] is the side containing a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the [latex]-x[\/latex] from the right side by adding [latex]x[\/latex] to both sides.<\/td>\r\n<td>[latex]7x\\color{red}{+x}=-x\\color{red}{+x}+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8x=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All the variables are on the left and the constants are on the right. Divide both sides by [latex]8[\/latex].<\/td>\r\n<td>[latex]\\frac{8x}{\\color{red}{8}}=\\frac{24}{\\color{red}{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]7x=-x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]x=3[\/latex].<\/td>\r\n<td>[latex]7(\\color{red}{3})\\stackrel{\\text{?}}{=}-(\\color{red}{3})+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nDid you see the subtle difference between the two equations? In the first, the right side looked like this: [latex]2x+7[\/latex], and in the second, the right side looked like this: [latex]-x+24[\/latex], even though they look different, we still used the same techniques to solve both.\r\n\r\nNow you can try solving an equation with variables on both sides where it is beneficial to move the variable term to the left side.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142129&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"220\"><\/iframe>\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142132&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"220\"><\/iframe>\r\n\r\n<\/div>\r\nIn our last examples, we moved the variable term to the left side of the equation. In the next example, you will see that it is beneficial to move the variable term to the right side of the equation. There is no \"correct\" side to move the variable term, but the choice can help you avoid working with negative signs.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]5y - 8=7y[\/latex].\r\n[reveal-answer q=\"100719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"100719\"]\r\n\r\nSolution:\r\nThe only constant, [latex]-8[\/latex], is on the left side of the equation, and the variable, [latex]y[\/latex], is on both sides. Let\u2019s leave the constant on the left and collect the variables to the right.\r\n<table id=\"eip-id1168468768462\" class=\"unnumbered unstyled\" summary=\"The first line says 5y minus 8 equals 7y. The left side is labeled \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]5y-8[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">constant<\/span>.<\/span>\r\n\r\n<span style=\"color: #000000;\">[latex]7y[\/latex] is the side containing only a <\/span><span style=\"color: #ff0000;\">variable<\/span><span style=\"color: #000000;\">.<\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]5y[\/latex] from both sides.<\/td>\r\n<td>[latex]5y\\color{red}{-5y}-8=7y\\color{red}{-5y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-8=2y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We have the variables on the right and the constants on the left. Divide both sides by [latex]2[\/latex].<\/td>\r\n<td>[latex]\\frac{-8}{\\color{red}{2}}=\\frac{2y}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-4=y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite with the variable on the left.<\/td>\r\n<td>[latex]y=-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]5y-8=7y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]y=-4[\/latex].<\/td>\r\n<td>[latex]5(\\color{red}{-4})-8\\stackrel{\\text{?}}{=}7(\\color{red}{-4})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-20-8\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-28=-28\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try solving an equation where it is beneficial to move the variable term to the right side.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142125&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Solve Equations with Variables and Constants on Both Sides<\/h3>\r\nThe next example will be the first to have variables <em>and<\/em> constants on both sides of the equation. As we did before, we\u2019ll collect the variable terms to one side and the constants to the other side. You will see that as the number of variable and constant terms increases, so do the number of steps it takes to solve the equation.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nSolve: [latex]7x+5=6x+2[\/latex].\r\n[reveal-answer q=\"859740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"859740\"]\r\n\r\nSolution:\r\nStart by choosing which side will be the variable side and which side will be the constant side. The variable terms are [latex]7x[\/latex] and [latex]6x[\/latex]. Since [latex]7[\/latex] is greater than [latex]6[\/latex], make the left side the variable side and so the right side will be the constant side.\r\n\r\n[latex]16=16\\quad\\checkmark[\/latex]\r\n<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]7x+5=6x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Collect the variable terms to the left side by subtracting [latex]6x[\/latex] from both sides.<\/td>\r\n<td>[latex]7x\\color{red}{-6x}+5=6x\\color{red}{-6x}+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x+5=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now, collect the constants to the right side by subtracting [latex]5[\/latex] from both sides.<\/td>\r\n<td>[latex]x+5\\color{red}{-5}=2\\color{red}{-5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The solution is [latex]x=-3[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]7x+5=6x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=-3[\/latex].<\/td>\r\n<td>[latex]7(\\color{red}{-3})+5\\stackrel{\\text{?}}{=}6(\\color{red}{-3})+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-21+5\\stackrel{\\text{?}}{=}-18+2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSolve: [latex]6n - 2=-3n+7[\/latex].\r\n\r\n[reveal-answer q=\"399032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399032\"]\r\nWe have [latex]6n[\/latex] on the left and [latex]-3n[\/latex] on the right. Since [latex]6&gt;-3[\/latex], make the left side the \"variable\" side.\r\n<table id=\"eip-id1168467335489\" class=\"unnumbered unstyled\" summary=\"The top line says 6n minus 2 equals negative 3n plus 7. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]6n-2=-3n+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We don't want variables on the right side\u2014add [latex]3n[\/latex] to both sides to leave only constants on the right.<\/td>\r\n<td>[latex]6n\\color{red}{+3n}-2=-3n\\color{red}{+3n}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]9n-2=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We don't want any constants on the left side, so add [latex]2[\/latex] to both sides.<\/td>\r\n<td>[latex]9n-2\\color{red}{+2}=7\\color{red}{+2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]9n=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The variable term is on the left and the constant term is on the right. To get the coefficient of [latex]n[\/latex] to be one, divide both sides by [latex]9[\/latex].<\/td>\r\n<td>[latex]\\frac{9n}{\\color{red}{9}}=\\frac{9}{\\color{red}{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]n=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]6n-2=-3n+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]1[\/latex] for [latex]n[\/latex].<\/td>\r\n<td>[latex]6(\\color{red}{1})-2\\stackrel{\\text{?}}{=}-3(\\color{red}{1})+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4=4\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show an example of how to solve a multi-step equation by moving the variable terms to one side and the constants to the other side. You will see that it doesn't matter which side you choose to be the variable side; you can get the correct answer either way.\r\n\r\nhttps:\/\/youtu.be\/_hBoWoctfAo\r\n\r\nIn the next example, we move the variable terms to the right side to keep a positive coefficient on the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]2a - 7=5a+8[\/latex].\r\n\r\n[reveal-answer q=\"654456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654456\"]\r\n\r\nSolution:\r\n\r\nThis equation has [latex]2a[\/latex] on the left and [latex]5a[\/latex] on the right. Since [latex]5&gt;2[\/latex], make the right side the variable side and the left side the constant side.\r\n\r\nLet [latex]a=-5[\/latex].[latex]2(\\color{red}{-5})-7\\stackrel{\\text{?}}{=}5(\\color{red}{-5})+8[\/latex]\r\n<table id=\"eip-id1168466004451\" class=\"unnumbered unstyled\" summary=\"The top line says 6n minus 2 equals negative 3n plus 7. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]2a-7=5a+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]2a[\/latex] from both sides to remove the variable term from the left.<\/td>\r\n<td>[latex]2a\\color{red}{-2a}-7=5a\\color{red}{-2a}+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]-7=3a+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]8[\/latex] from both sides to remove the constant from the right.<\/td>\r\n<td>[latex]-7\\color{red}{-8}=3a+8\\color{red}{-8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-15=3a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by [latex]3[\/latex] to make 1[\/latex] the coefficient of [latex]a[\/latex] .<\/td>\r\n<td>[latex]\\frac{-15}{\\color{red}{3}}=\\frac{3a}{\\color{red}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-5=a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]2a-7=5a+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-10-7\\stackrel{\\text{?}}{=}-25+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-17=-17\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of solving a multi-step\u00a0equation by moving the variable terms to one side and the constants to the other side.\r\n\r\nhttps:\/\/youtu.be\/kiYPW6hrTS4\r\n\r\nTry these problems to see how well you understand how to solve linear equations with variables and constants on both sides of the equal sign.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142134&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142136&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nWe just showed a lot of examples of different kinds of linear equations you may encounter. There are some good habits to develop that will help you solve all kinds of linear equations. We\u2019ll summarize the steps we took so you can easily refer to them.\r\n<div class=\"textbox shaded\">\r\n<h3>Solve an equation with variables and constants on both sides<\/h3>\r\n<ol id=\"eip-id1168468371331\" class=\"stepwise\">\r\n \t<li>Choose one side to be the variable side and then the other will be the constant side.<\/li>\r\n \t<li>Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Collect the constants to the other side, using the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Make the coefficient of the variable [latex]1[\/latex], using the Multiplication or Division Property of Equality.<\/li>\r\n \t<li>Check the solution by substituting it into the original equation.<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether a number is a solution to an equation<\/li>\n<li>Review the use of the subtraction and addition properties of equality to solve linear equations<\/li>\n<li>Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality<\/li>\n<li>Check your solution to a linear equation to verify its accuracy<\/li>\n<li>Review and use the division and multiplication properties of equality to solve linear equations<\/li>\n<li>Use a reciprocal to solve a linear equation that contains fractions<\/li>\n<li>Solve a linear equation that requires simplification before using properties of equality<\/li>\n<li>Solve a linear equation that requires a combination of the properties of equality<\/li>\n<li>Solve linear equations by isolating constants and variables<\/li>\n<li>Solve linear equations with variables on both sides that require several steps<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Using the Subtraction and Addition Properties for Single-Step Equations<\/h2>\n<p>We began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Solution of an Equation<\/h3>\n<p>A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p>\n<p>In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.<\/p>\n<p>Determine whether a number is a solution to an equation.<\/p>\n<ol id=\"eip-95\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"fs-id1703984\">\n<li>If it is true, the number is a solution.<\/li>\n<li>If it is not true, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Determine whether [latex]y=\\frac{3}{4}[\/latex] is a solution for [latex]4y+3=8y[\/latex].<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466426761\" class=\"unnumbered unstyled\" summary=\"The top line says 4y plus 3 equals 8y. Beside this is\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]4y+3=8y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{\\frac{3}{4}}[\/latex] for [latex]y[\/latex]<\/td>\n<td>[latex]4(\\color{red}{\\frac{3}{4}})+3\\stackrel{\\text{?}}{=}8(\\color{red}{\\frac{3}{4}})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+3\\stackrel{\\text{?}}{=}6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]6=6\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]y=\\frac{3}{4}[\/latex] results in a true equation, [latex]\\frac{3}{4}[\/latex] is a solution to the equation [latex]4y+3=8y[\/latex].<\/p>\n<\/div>\n<p>Now it is your turn to determine whether a fraction is the solution to an equation.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141717&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"380\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141719&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"380\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1956928\"><strong>Subtraction Property of Equality<\/strong><\/h3>\n<p>For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1166490863495\"><strong>Addition Property of Equality<\/strong><\/h3>\n<p>For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/p>\n<\/div>\n<p>The goal is to isolate the variable on one side of the equation.<\/p>\n<p>Some people picture a balance scale, as in the image below, when they solve equations.<\/p>\n<figure id=\"CNX_BMath_Figure_08_01_001\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222535\/CNX_BMath_Figure_08_01_001.png\" alt=\"Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is\" \/><\/figure>\n<p>The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.<\/p>\n<p>&nbsp;<\/p>\n<p>In the following example we\u00a0review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation to make sure you have a true statement.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]x+11=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nTo isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.<\/p>\n<table id=\"eip-id1168468291100\" class=\"unnumbered unstyled\" summary=\"The top line says x plus 11 equals negative 3. The next line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]x+11=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Subtract 11 from each side to &#8220;undo&#8221; the addition.<\/td>\n<td>[latex]x+11\\color{red}{-11}=-3\\color{red}{-11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]x=-14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]x+11=-3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]x=-14[\/latex] .<\/td>\n<td>[latex]\\color{red}{-14}+11\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try solving an equation that requires using the addition property.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141721&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the original equation in the previous example, [latex]11[\/latex] was added to the [latex]x[\/latex] , so we subtracted [latex]11[\/latex] to &#8220;undo&#8221; the addition. In the next example, we will need to &#8220;undo&#8221; subtraction by using the Addition Property of Equality.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]m - 4=-5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604060\">Show Solution<\/span><\/p>\n<div id=\"q604060\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467157298\" class=\"unnumbered unstyled\" summary=\"The first line says m minus 4 equals negative 5. The next line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]m-4=-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add 4 to each side to &#8220;undo&#8221; the subtraction.<\/td>\n<td>[latex]m-4\\color{red}{+4}=-5\\color{red}{+4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]m=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]m-4=-5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]m=-1[\/latex] .<\/td>\n<td>[latex]\\color{red}{-1}+4\\stackrel{\\text{?}}{=}-5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5=-5\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>The solution to [latex]m - 4=-5[\/latex] is [latex]m=-1[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try using the addition property to solve an equation.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom27\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141723&amp;theme=oea&amp;iframe_resize_id=mom27\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we present more examples of solving equations using the addition and subtraction properties.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yqdlj0lv7Cc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Subtraction and Addition Properties for Multi-Step Equations<\/h2>\n<p>In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve:<\/p>\n<p>[latex]3x - 7 - 2x - 4=1[\/latex].<\/p>\n<p>Solution:<br \/>\nThe left side of the equation has an expression that we should simplify before trying to isolate the variable.<\/p>\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]x-11=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\n<td>[latex]x-11\\color{red}{+11}=1\\color{red}{+11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/p>\n<p>Substitute [latex]x=12[\/latex] into the original equation.<br \/>\n[latex]3x-7-2x-4=1[\/latex]<\/p>\n<p>[latex]3(\\color{red}{12})-7-2(\\color{red}{12})-4=1[\/latex]<\/p>\n<p>[latex]36-7-24-4=1[\/latex]<\/p>\n<p>[latex]29-24-4=1[\/latex]<\/p>\n<p>[latex]5-4=1[\/latex]<\/p>\n<p>[latex]1=1\\quad\\checkmark[\/latex]<\/p>\n<p>The solution checks.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now you can try solving a couple\u00a0equations where you should simplify first.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The last few examples involved simplifying using addition and subtraction. Let&#8217;s look at an example where we need to distribute first in order to simplify the equation as much as possible.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]3\\left(n - 4\\right)-2n=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe left side of the equation has an expression that we should simplify.<\/p>\n<table id=\"eip-id1168468254328\" class=\"unnumbered unstyled\" summary=\"The top line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3(n-4)-2n=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute on the left.<\/td>\n<td>[latex]3n-12-2n=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to rearrange terms.<\/td>\n<td>[latex]3n-2n-12=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]n-12=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Isolate <em>n<\/em> using the Addition Property of Equality.<\/td>\n<td>[latex]n-12\\color{red}{+12}=-3\\color{red}{+12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]n=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/p>\n<p>Substitute [latex]n=9[\/latex] into the original equation.<br \/>\n[latex]3(n-4)-2n=-3[\/latex]<br \/>\n[latex]3(\\color{red}{9}-4)-2\\cdot\\color{red}{9}=-3[\/latex]<br \/>\n[latex]3(5)-18=-3[\/latex]<br \/>\n[latex]15-18=-3[\/latex]<br \/>\n[latex]-3=-3\\quad\\checkmark[\/latex]<br \/>\nThe solution checks.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a few problems that involve distribution.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141737&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The next example has expressions on both sides that need to be simplified.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex].<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190976\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q190976\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nBoth sides of the equation have expressions that we should simplify before we isolate the variable.<\/p>\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute on the left, subtract on the right.<\/td>\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property of Addition.<\/td>\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\n<td>[latex]k-2\\color{red}{+2}=-9\\color{red}{+2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]k=-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/p>\n<p>Let [latex]k=-7[\/latex].<br \/>\n[latex]2(3k-1)-5k=-2-7[\/latex]<br \/>\n[latex]2(3(\\color{red}{-7}-1)-5(\\color{red}{-7})=-2-7[\/latex]<br \/>\n[latex]2(-21-1)-5(-7)=-9[\/latex]<br \/>\n[latex]2(-22)+35=-9[\/latex]<br \/>\n[latex]-44+35=-9[\/latex]<br \/>\n[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The solution checks.<\/p>\n<p class=\"p1\"><\/div>\n<\/div>\n<\/div>\n<p>Now, you give it a try!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom220\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141739&amp;theme=oea&amp;iframe_resize_id=mom220\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solve Linear Equations in One Variable with Simplifying (One-Step Add\/Subtract)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/shGKzDBA5kQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Division and Multiplication Properties of Equality for Single-Step Equations<\/h2>\n<p>Let&#8217;s review the Division and Multiplication Properties of Equality as we prepare to use them to solve single-step equations.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Division Property of Equality<\/h3>\n<p>For all real numbers [latex]a,b,c[\/latex], and [latex]c\\ne 0[\/latex], if [latex]a=b[\/latex], then [latex]\\frac{a}{c}=\\frac{b}{c}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Multiplication Property of Equality<\/h3>\n<p>For all real numbers [latex]a,b,c[\/latex], if [latex]a=b[\/latex], then [latex]ac=bc[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Stated simply, when you divide or multiply both sides of an equation by the same quantity, you still have equality.<\/p>\n<p>Let\u2019s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to &#8220;undo&#8221; the operation on the variable. In the example below the variable is multiplied by [latex]4[\/latex], so we will divide both sides by [latex]4[\/latex] to &#8220;undo&#8221; the multiplication.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]4x=-28[\/latex].<\/p>\n<p>Solution:<\/p>\n<p>To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]4x=-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by 4 to undo the multiplication.<\/td>\n<td>[latex]\\frac{4x}{\\color{red}4}=\\frac{-28}{\\color{red}4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x =-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer.<\/td>\n<td>[latex]4x=-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=-7[\/latex]. Substitute [latex]-7[\/latex] for x.<\/td>\n<td>[latex]4(\\color{red}{-7})\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]-28=-28[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since this is a true statement, [latex]x=-7[\/latex] is a solution to [latex]4x=-28[\/latex].<\/p>\n<\/div>\n<p>Now you can try to solve an equation that requires division and\u00a0includes negative numbers.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141857&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the previous example, to &#8220;undo&#8221; multiplication, we divided. How do you think we &#8220;undo&#8221; division? Next, we will show an example that requires us to use multiplication to undo division.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]\\frac{a}{-7}=-42[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q399032\">Show Solution<\/span><\/p>\n<div id=\"q399032\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nHere [latex]a[\/latex] is divided by [latex]-7[\/latex]. We can multiply both sides by [latex]-7[\/latex] to isolate [latex]a[\/latex].<\/p>\n<table id=\"eip-id1168468288515\" class=\"unnumbered unstyled\" summary=\"The top shows a over negative 7 equals negative 42. The next line says\">\n<tbody>\n<tr>\n<td>[latex]\\frac{a}{-7}=-42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply both sides by [latex]-7[\/latex] .<\/td>\n<td>[latex]\\color{red}{-7}(\\frac{a}{-7})=\\color{red}{-7}(-42)[\/latex]<\/p>\n<p>[latex]\\frac{-7a}{-7}=294[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]a=294[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer.<\/td>\n<td>[latex]\\frac{a}{-7}=-42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]a=294[\/latex] .<\/td>\n<td>[latex]\\frac{\\color{red}{294}}{-7}\\stackrel{\\text{?}}{=}-42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-42=-42\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now see if you can solve a\u00a0problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers &#8211; two negatives give a positive when they are multiplied.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom21\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141868&amp;theme=oea&amp;iframe_resize_id=mom21\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable. \u00a0As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]-r=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388033\">Show Solution<\/span><\/p>\n<div id=\"q388033\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nRemember [latex]-r[\/latex] is equivalent to [latex]-1r[\/latex].<\/p>\n<table id=\"eip-id1168469604717\" class=\"unnumbered unstyled\" summary=\"The first line says negative r equals 2. The next line says\">\n<tbody>\n<tr>\n<td>[latex]-r=2[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Rewrite [latex]-r[\/latex] as [latex]-1r[\/latex] .<\/td>\n<td>[latex]-1r=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by [latex]-1[\/latex] .<\/td>\n<td>[latex]\\frac{-1r}{\\color{red}{-1}}=\\frac{2}{\\color{red}{-1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]r=-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/td>\n<td>[latex]-r=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]r=-2[\/latex]<\/td>\n<td>[latex]-(\\color{red}{-2})\\stackrel{\\text{?}}{=}2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2=2\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try to solve an equation with a negative variable.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141865&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]\\frac{2}{3}x=18[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q444022\">Show Solution<\/span><\/p>\n<div id=\"q444022\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nSince the product of a number and its reciprocal is [latex]1[\/latex], our strategy will be to isolate [latex]x[\/latex] by multiplying by the reciprocal of [latex]\\frac{2}{3}[\/latex].<\/p>\n<table id=\"eip-id1168468646645\" class=\"unnumbered unstyled\" summary=\"The first line shows two-thirds x equals 18. The next line says\">\n<tbody>\n<tr style=\"height: 37px;\">\n<td style=\"height: 37px;\">[latex]\\frac{2}{3}x=18[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 38px;\">\n<td style=\"height: 38px;\">Multiply by the reciprocal of [latex]\\frac{2}{3}[\/latex] .<\/td>\n<td style=\"height: 38px;\">[latex]\\frac{\\color{red}{3}}{\\color{red}{2}}\\cdot\\frac{2}{3}x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 37px;\">\n<td style=\"height: 37px;\">Reciprocals multiply to one.<\/td>\n<td style=\"height: 37px;\">[latex]1x=\\frac{3}{2}\\cdot\\frac{18}{1}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px;\">Multiply.<\/td>\n<td style=\"height: 18px;\">[latex]x=27[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Check your answer.<\/td>\n<td style=\"height: 44.8594px;\">[latex]\\frac{2}{3}x=18[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44.8594px;\">\n<td style=\"height: 14px;\">Let [latex]x=27[\/latex].<\/td>\n<td style=\"height: 45px;\">[latex]\\frac{2}{3}\\cdot\\color{red}{27}\\stackrel{\\text{?}}{=}18[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td><\/td>\n<td style=\"height: 24px;\">[latex]18=18\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that we could have divided both sides of the equation [latex]\\frac{2}{3}x=18[\/latex] by [latex]\\frac{2}{3}[\/latex] to isolate [latex]x[\/latex]. While this would work, multiplying by the reciprocal requires fewer steps.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom22\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141871&amp;theme=oea&amp;iframe_resize_id=mom22\" width=\"100%\" height=\"260\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Solving One Step Equation by Mult\/Div.  Integers (Var on Right)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TB1rkPbF8rA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Division and Multiplication Properties of Equality for Multi-Step Equations<\/h2>\n<p>Many equations start out more complicated than the ones we\u2019ve just solved. Our goal has been to familiarize you with the many ways to apply the addition, subtraction, multiplication, and division properties that are used to solve equations algebraically. Let&#8217;s work through an example that will employ the following techniques:<\/p>\n<ul>\n<li>simplify by combining like terms<\/li>\n<li>isolate x by using the division property of equality<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve: [latex]8x+9x - 5x=-3+15[\/latex].<\/p>\n<p>Solution:<\/p>\n<p>First, we need to simplify both sides of the equation as much as possible<\/p>\n<p>Start by combining like terms to simplify each side.<\/p>\n<table id=\"eip-id1168466098204\" class=\"unnumbered unstyled\" summary=\"The first line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8x+9x-5x=-3+15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]12x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by 12 to isolate x.<\/td>\n<td>[latex]\\frac{12x}{\\color{red}{12}}=\\frac{12}{\\color{red}{12}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer. Let [latex]x=1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]8x+9x-5x=-3+15[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]8\\cdot\\color{red}{1}+9\\cdot\\color{red}{1}-5\\cdot\\color{red}{1}\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]8+9-5\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]12=12\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Here is a similar problem for you to try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141884&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>You may not always have the variables on the left side of the equation, so we will show an example with variables on the right side. You will see that the properties used to solve this equation are exactly the same as the previous example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]11 - 20=17y - 8y - 6y[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q399032\">Show Solution<\/span><\/p>\n<div id=\"q399032\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<p>Simplify each side by combining like terms.<\/p>\n<table id=\"eip-id1168466111452\" class=\"unnumbered unstyled\" summary=\"The first line shows 11 minus 20 equals 17y minus 8y minus 6y. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]11-20=17y-8y-6y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each side.<\/td>\n<td>[latex]-9=3y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by 3 to isolate y.<\/td>\n<td>[latex]\\frac{-9}{\\color{red}{3}}=\\frac{3y}{\\color{red}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-3=y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer. Let [latex]y=-3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11-20=17y-8y-6y[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11-20\\stackrel{\\text{?}}{=}17(  \\color{red}{-3})-8(\\color{red}{-3})-6(\\color{red}{-3})[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11-20\\stackrel{\\text{?}}{=}-51+24+18[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.<\/p>\n<p>Now you can try solving a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom23\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141901&amp;theme=oea&amp;iframe_resize_id=mom23\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In our next example, we have an equation that contains a set of parentheses. \u00a0We will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]-3\\left(n - 2\\right)-6=21[\/latex].<\/p>\n<p>Remember\u2014always simplify each side first.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q789987\">Show Solution<\/span><\/p>\n<div id=\"q789987\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468278405\" class=\"unnumbered unstyled\" summary=\"The first line shows negative 3 times parentheses n minus 2 minus 6 equals 21. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]-3n+6-6=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-3n=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by -3 to isolate n.<\/td>\n<td>[latex]\\frac{-3n}{\\color{red}{-3}}=\\frac{21}{\\color{red}{-3}}[\/latex]<\/p>\n<p>[latex]n=-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer. Let [latex]n=-7[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(\\color{red}{-7}-2)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(-9)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]27-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom27\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=141911&amp;theme=oea&amp;iframe_resize_id=mom27\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In the following\u00a0video you will see another example of using the division property of equality to solve an equation as well as \u00a0another example of how to solve a multi-step equation that includes a set of parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Solve Linear Equations in One Variable with Simplifying (One-Step Mult\/Div)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qe89pkRKzRw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Solving Equations With Variables on Both Sides<\/h2>\n<p>The equations we solved in the last section\u00a0simplified nicely so that we could use the division property to isolate the variable and solve the equation. Sometimes, after you simplify you may have a variable and a constant term on the same side of the equal sign.<\/p>\n<p>Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate\u00a0the variable terms on one side of the equation.<\/p>\n<p>Read on to find out how to solve this kind of equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Solve: [latex]4x+6=-14[\/latex].<\/p>\n<p>Solution:<\/p>\n<p>In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.<\/p>\n<table style=\"width: 70%;\" summary=\"The top line says 4x plus 6 equals negative 14.\">\n<tbody>\n<tr style=\"height: 45.8594px;\">\n<td style=\"height: 45.8594px; width: 1179.02px;\" colspan=\"2\">Since the left side is the variable side, the 6 is out of place. We must &#8220;undo&#8221; adding [latex]6[\/latex] by subtracting [latex]6[\/latex], and to keep the equality we must subtract [latex]6[\/latex] from both sides. Use the Subtraction Property of Equality.<\/td>\n<td style=\"height: 45.8594px; width: 176px;\">[latex]4x+6\\color{red}{-6}=-14\\color{red}{-6}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]4x=-20[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Now all the [latex]x[\/latex] s are on the left and the constant on the right.<\/td>\n<td style=\"height: 15px; width: 176px;\"><\/td>\n<\/tr>\n<tr style=\"height: 60px;\">\n<td style=\"height: 60px; width: 1179.02px;\" colspan=\"2\">Use the Division Property of Equality.<\/td>\n<td style=\"height: 60px; width: 176px;\">[latex]\\frac{4x}{\\color{red}{4}}=\\frac{-20}{\\color{red}{4}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]x=-5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Check:<\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]4x+6=-14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\">Let [latex]x=-5[\/latex] .<\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]4(\\color{red}{-5})+6=-14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]-20+6=-14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15px; width: 176px;\">[latex]-14=-14\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Solve: [latex]2y - 7=15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q629971\">Show Solution<\/span><\/p>\n<div id=\"q629971\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nNotice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the [latex]7[\/latex] is out of place. It is subtracted from the [latex]2y[\/latex], so to &#8220;undo&#8221; subtraction, add [latex]7[\/latex] to both sides.<\/p>\n<table id=\"eip-id1168469592645\" class=\"unnumbered unstyled\" summary=\"The first line says 2y minus 7 equals 15. The left side is labeled\">\n<tbody>\n<tr>\n<td>[latex]2y-7[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><\/p>\n<p><span style=\"color: #000000;\">[latex]15[\/latex] is the side containing only a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add [latex]7[\/latex] to both sides.<\/td>\n<td>[latex]2y-7\\color{red}{+7}=15\\color{red}{+7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]2y=22[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\">The variables are now on one side and the constants on the other.<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Divide both sides by [latex]2[\/latex].<\/td>\n<td>[latex]\\frac{2y}{\\color{red}{2}}=\\frac{22}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]y=11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Check:<\/td>\n<td>[latex]2y-7=15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Let [latex]y=11[\/latex] .<\/td>\n<td>[latex]2\\cdot\\color{red}{11}-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]22-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]15=15\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try\u00a0a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try\u00a0It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142131&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Solve Equations with Variables on Both Sides<\/h3>\n<p>You may have noticed that in all the equations we have solved so far, we had\u00a0variables on only one\u00a0side of the equation. This does not happen all the time\u2014so now we\u2019ll see how to solve equations where there are variable terms on both sides of the equation. We will start like we did above\u2014choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side as well.<\/p>\n<p>In the next example,\u00a0the variable, [latex]x[\/latex], is on both sides, but the constants appear only on the right side, so we&#8217;ll make the right side the &#8220;constant&#8221; side. Then the left side will be the &#8220;variable&#8221; side.<\/p>\n<div class=\"textbox exercises\">\n<h3>ExampleS<\/h3>\n<p>Solve: [latex]5x=4x+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q235739\">Show Solution<\/span><\/p>\n<div id=\"q235739\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168465095460\" class=\"unnumbered unstyled\" summary=\"The first line says 5x equals 4x plus 7. The left side is labeled\">\n<tbody>\n<tr>\n<td colspan=\"2\">[latex]5x[\/latex] is the side containing only a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><span style=\"color: #000000;\">[latex]4x+7[\/latex] is the side containing a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">We don&#8217;t want any variables on the right, so subtract the [latex]4x[\/latex] .<\/td>\n<td>[latex]5x\\color{red}{-4x}=4x\\color{red}{-4x}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]x=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">We have all the variables on one side and the constants on the other. We have solved the equation.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td><\/td>\n<td>[latex]5x=4x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]7[\/latex] for [latex]x[\/latex] .<\/td>\n<td><\/td>\n<td>[latex]5(\\color{red}{7})\\stackrel{\\text{?}}{=}4(\\color{red}{7})+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>[latex]35\\stackrel{\\text{?}}{=}28+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>[latex]35=35\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Solve: [latex]7x=-x+24[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q192799\">Show Solution<\/span><\/p>\n<div id=\"q192799\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe only constant, [latex]24[\/latex], is on the right, so let the left side be the variable side.<\/p>\n<table id=\"eip-id1168467377760\" class=\"unnumbered unstyled\" summary=\"The first line says 7x equals negative x plus 24. The left side is labeled\">\n<tbody>\n<tr>\n<td>[latex]7x[\/latex] is the side containing only a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><\/p>\n<p><span style=\"color: #000000;\">[latex]-x+24[\/latex] is the side containing a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Remove the [latex]-x[\/latex] from the right side by adding [latex]x[\/latex] to both sides.<\/td>\n<td>[latex]7x\\color{red}{+x}=-x\\color{red}{+x}+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8x=24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All the variables are on the left and the constants are on the right. Divide both sides by [latex]8[\/latex].<\/td>\n<td>[latex]\\frac{8x}{\\color{red}{8}}=\\frac{24}{\\color{red}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]7x=-x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]x=3[\/latex].<\/td>\n<td>[latex]7(\\color{red}{3})\\stackrel{\\text{?}}{=}-(\\color{red}{3})+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Did you see the subtle difference between the two equations? In the first, the right side looked like this: [latex]2x+7[\/latex], and in the second, the right side looked like this: [latex]-x+24[\/latex], even though they look different, we still used the same techniques to solve both.<\/p>\n<p>Now you can try solving an equation with variables on both sides where it is beneficial to move the variable term to the left side.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142129&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"220\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142132&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"220\"><\/iframe><\/p>\n<\/div>\n<p>In our last examples, we moved the variable term to the left side of the equation. In the next example, you will see that it is beneficial to move the variable term to the right side of the equation. There is no &#8220;correct&#8221; side to move the variable term, but the choice can help you avoid working with negative signs.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]5y - 8=7y[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100719\">Show Solution<\/span><\/p>\n<div id=\"q100719\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe only constant, [latex]-8[\/latex], is on the left side of the equation, and the variable, [latex]y[\/latex], is on both sides. Let\u2019s leave the constant on the left and collect the variables to the right.<\/p>\n<table id=\"eip-id1168468768462\" class=\"unnumbered unstyled\" summary=\"The first line says 5y minus 8 equals 7y. The left side is labeled\">\n<tbody>\n<tr>\n<td>[latex]5y-8[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/p>\n<p><span style=\"color: #000000;\">[latex]7y[\/latex] is the side containing only a <\/span><span style=\"color: #ff0000;\">variable<\/span><span style=\"color: #000000;\">.<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]5y[\/latex] from both sides.<\/td>\n<td>[latex]5y\\color{red}{-5y}-8=7y\\color{red}{-5y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-8=2y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We have the variables on the right and the constants on the left. Divide both sides by [latex]2[\/latex].<\/td>\n<td>[latex]\\frac{-8}{\\color{red}{2}}=\\frac{2y}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-4=y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite with the variable on the left.<\/td>\n<td>[latex]y=-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]5y-8=7y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]y=-4[\/latex].<\/td>\n<td>[latex]5(\\color{red}{-4})-8\\stackrel{\\text{?}}{=}7(\\color{red}{-4})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-20-8\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-28=-28\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try solving an equation where it is beneficial to move the variable term to the right side.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142125&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Solve Equations with Variables and Constants on Both Sides<\/h3>\n<p>The next example will be the first to have variables <em>and<\/em> constants on both sides of the equation. As we did before, we\u2019ll collect the variable terms to one side and the constants to the other side. You will see that as the number of variable and constant terms increases, so do the number of steps it takes to solve the equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Solve: [latex]7x+5=6x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q859740\">Show Solution<\/span><\/p>\n<div id=\"q859740\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nStart by choosing which side will be the variable side and which side will be the constant side. The variable terms are [latex]7x[\/latex] and [latex]6x[\/latex]. Since [latex]7[\/latex] is greater than [latex]6[\/latex], make the left side the variable side and so the right side will be the constant side.<\/p>\n<p>[latex]16=16\\quad\\checkmark[\/latex]<\/p>\n<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says\">\n<tbody>\n<tr>\n<td>[latex]7x+5=6x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Collect the variable terms to the left side by subtracting [latex]6x[\/latex] from both sides.<\/td>\n<td>[latex]7x\\color{red}{-6x}+5=6x\\color{red}{-6x}+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x+5=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Now, collect the constants to the right side by subtracting [latex]5[\/latex] from both sides.<\/td>\n<td>[latex]x+5\\color{red}{-5}=2\\color{red}{-5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The solution is [latex]x=-3[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]7x+5=6x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=-3[\/latex].<\/td>\n<td>[latex]7(\\color{red}{-3})+5\\stackrel{\\text{?}}{=}6(\\color{red}{-3})+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-21+5\\stackrel{\\text{?}}{=}-18+2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Solve: [latex]6n - 2=-3n+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q399032\">Show Solution<\/span><\/p>\n<div id=\"q399032\" class=\"hidden-answer\" style=\"display: none\">\nWe have [latex]6n[\/latex] on the left and [latex]-3n[\/latex] on the right. Since [latex]6>-3[\/latex], make the left side the &#8220;variable&#8221; side.<\/p>\n<table id=\"eip-id1168467335489\" class=\"unnumbered unstyled\" summary=\"The top line says 6n minus 2 equals negative 3n plus 7. The next line says\">\n<tbody>\n<tr>\n<td>[latex]6n-2=-3n+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We don&#8217;t want variables on the right side\u2014add [latex]3n[\/latex] to both sides to leave only constants on the right.<\/td>\n<td>[latex]6n\\color{red}{+3n}-2=-3n\\color{red}{+3n}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]9n-2=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We don&#8217;t want any constants on the left side, so add [latex]2[\/latex] to both sides.<\/td>\n<td>[latex]9n-2\\color{red}{+2}=7\\color{red}{+2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]9n=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The variable term is on the left and the constant term is on the right. To get the coefficient of [latex]n[\/latex] to be one, divide both sides by [latex]9[\/latex].<\/td>\n<td>[latex]\\frac{9n}{\\color{red}{9}}=\\frac{9}{\\color{red}{9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]n=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]6n-2=-3n+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]1[\/latex] for [latex]n[\/latex].<\/td>\n<td>[latex]6(\\color{red}{1})-2\\stackrel{\\text{?}}{=}-3(\\color{red}{1})+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4=4\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show an example of how to solve a multi-step equation by moving the variable terms to one side and the constants to the other side. You will see that it doesn&#8217;t matter which side you choose to be the variable side; you can get the correct answer either way.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Solve a Linear Equation in One Variable with Variables on Both Sides: 2x+8=-2x-24\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_hBoWoctfAo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we move the variable terms to the right side to keep a positive coefficient on the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]2a - 7=5a+8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654456\">Show Solution<\/span><\/p>\n<div id=\"q654456\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<p>This equation has [latex]2a[\/latex] on the left and [latex]5a[\/latex] on the right. Since [latex]5>2[\/latex], make the right side the variable side and the left side the constant side.<\/p>\n<p>Let [latex]a=-5[\/latex].[latex]2(\\color{red}{-5})-7\\stackrel{\\text{?}}{=}5(\\color{red}{-5})+8[\/latex]<\/p>\n<table id=\"eip-id1168466004451\" class=\"unnumbered unstyled\" summary=\"The top line says 6n minus 2 equals negative 3n plus 7. The next line says\">\n<tbody>\n<tr>\n<td>[latex]2a-7=5a+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]2a[\/latex] from both sides to remove the variable term from the left.<\/td>\n<td>[latex]2a\\color{red}{-2a}-7=5a\\color{red}{-2a}+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]-7=3a+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]8[\/latex] from both sides to remove the constant from the right.<\/td>\n<td>[latex]-7\\color{red}{-8}=3a+8\\color{red}{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-15=3a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by [latex]3[\/latex] to make 1[\/latex] the coefficient of [latex]a[\/latex] .<\/td>\n<td>[latex]\\frac{-15}{\\color{red}{3}}=\\frac{3a}{\\color{red}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-5=a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]2a-7=5a+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-10-7\\stackrel{\\text{?}}{=}-25+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-17=-17\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of solving a multi-step\u00a0equation by moving the variable terms to one side and the constants to the other side.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Solve a Linear Equation in One Variable with Variables on Both Sides: 2m-9=6m-17\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/kiYPW6hrTS4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Try these problems to see how well you understand how to solve linear equations with variables and constants on both sides of the equal sign.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142134&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=142136&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>We just showed a lot of examples of different kinds of linear equations you may encounter. There are some good habits to develop that will help you solve all kinds of linear equations. We\u2019ll summarize the steps we took so you can easily refer to them.<\/p>\n<div class=\"textbox shaded\">\n<h3>Solve an equation with variables and constants on both sides<\/h3>\n<ol id=\"eip-id1168468371331\" class=\"stepwise\">\n<li>Choose one side to be the variable side and then the other will be the constant side.<\/li>\n<li>Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect the constants to the other side, using the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable [latex]1[\/latex], using the Multiplication or Division Property of Equality.<\/li>\n<li>Check the solution by substituting it into the original equation.<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":264444,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18213","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18213\/revisions"}],"predecessor-version":[{"id":18699,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18213\/revisions\/18699"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18213\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18213"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18213"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18213"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}