{"id":369,"date":"2018-04-17T02:32:37","date_gmt":"2018-04-17T02:32:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=369"},"modified":"2024-04-26T22:05:07","modified_gmt":"2024-04-26T22:05:07","slug":"solving-multi-step-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/solving-multi-step-equations\/","title":{"raw":"Solving Multi-Step Equations","rendered":"Solving Multi-Step Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning outcome<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve multi-step equations with variables on both sides&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Solve multi-step equations with variables on both sides<\/span><\/li>\r\n<\/ul>\r\n<\/div>\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 real life will take more steps to solve. Often, 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 \u2014 this is called combining like-terms.\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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nThe 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 all the techniques you've learned.\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]\\Large\\frac{12x}{\\color{red}{12}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141884&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\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\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 Answer[\/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]\\Large\\frac{-9}{\\color{red}{3}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.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\r\nLet's review how the Distributive Property works.\r\n<div class=\"textbox shaded\">\r\n<h3>Distributive Property<\/h3>\r\n<p style=\"padding-left: 30px;\">If [latex]a,b,c[\/latex] are real numbers, then<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]3\\left(x+4\\right)[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nSome students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/>\r\n<p style=\"text-align: center;\">[latex]3\\cdot x+3\\cdot 4[\/latex]<\/p>\r\nNow you give it a try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146473[\/ohm_question]\r\n\r\n<\/div>\r\nApply your understanding of the Distributive Property to an equation that needs to be solved.\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 Answer[\/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]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.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\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 Answer[\/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>\r\n\r\n<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 Answer[\/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]\\Large\\frac{8x}{\\color{red}{8}}\\normalsize =\\Large\\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\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:\/\/ohm.lumenlearning.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:\/\/ohm.lumenlearning.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 Answer[\/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]\\Large\\frac{-8}{\\color{red}{2}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.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<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 Answer[\/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<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" style=\"height: 195px;\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]7x+5=6x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Collect the variable terms to the left side by subtracting [latex]6x[\/latex] from both sides.<\/td>\r\n<td style=\"height: 30px;\">[latex]7x\\color{red}{-6x}+5=6x\\color{red}{-6x}+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]x+5=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Now, collect the constants to the right side by subtracting [latex]5[\/latex] from both sides.<\/td>\r\n<td style=\"height: 30px;\">[latex]x+5\\color{red}{-5}=2\\color{red}{-5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]x=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">The solution is [latex]x=-3[\/latex] .<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Check:<\/td>\r\n<td style=\"height: 15px;\">[latex]7x+5=6x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Let [latex]x=-3[\/latex].<\/td>\r\n<td style=\"height: 30px;\">[latex]7(\\color{red}{-3})+5\\stackrel{\\text{?}}{=}6(\\color{red}{-3})+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]-21+5\\stackrel{\\text{?}}{=}-18+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]16=16\\quad\\checkmark[\/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=\"399033\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"399033\"]\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><\/td>\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]\\Large\\frac{9n}{\\color{red}{9}}\\normalsize =\\Large\\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 Answer[\/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<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><\/td>\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 [latex]1[\/latex] the coefficient of [latex]a[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{-15}{\\color{red}{3}}\\normalsize =\\Large\\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>Let [latex]a=-5[\/latex]<\/td>\r\n<td>[latex]2(\\color{red}{-5})-7\\stackrel{\\text{?}}{=}5(\\color{red}{-5})+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:\/\/ohm.lumenlearning.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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142136&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nHere is a much more complex multi-step equation.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]\\Large{\\frac{x+22}{3}}=2x+4[\/latex]\r\n<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" style=\"height: 195px;\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{x+22}{3}=2x+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Multiply both sides of the equation by 3<\/td>\r\n<td style=\"height: 30px;\">[latex]\\color{red}{3}\\cdot\\frac{x+22}{3}=(2x+4)\\color{red}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Eliminate the [latex]\\frac{3}{3} = 1[\/latex] on the left and Distribute on the right<\/td>\r\n<td style=\"height: 15px;\">[latex]x+22=\\color{red}{3}(2x)+\\color{red}{3}(4)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Multiply<\/td>\r\n<td style=\"height: 30px;\">[latex]x+22=6x+12[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Subtract [latex]x[\/latex] from both sides (keeps variable term positive)<\/td>\r\n<td style=\"height: 30px;\">[latex]22=5x+12[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Subtract [latex]12[\/latex] from both sides (isolates the variable term)<\/td>\r\n<td style=\"height: 30px;\">[latex]10=5x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Divide both sides by [latex]5[\/latex] (isolates the variable)<\/td>\r\n<td style=\"height: 30px;\">[latex]2=x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Rewrite solution with variable on the right<\/td>\r\n<td style=\"height: 30px;\">[latex]x=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">The solution is [latex]x=2[\/latex]<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Check:<\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{x+22}{3}=2x+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Let [latex]x=-11[\/latex].<\/td>\r\n<td style=\"height: 30px;\">[latex]\\frac{\\color{red}{2}+22}{3}=2(\\color{red}{2})+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]\\frac{24}{3}\\stackrel{\\text{?}}{=}4+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]8=8\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\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 outcome<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve multi-step equations with variables on both sides&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Solve multi-step equations with variables on both sides<\/span><\/li>\n<\/ul>\n<\/div>\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 real life will take more steps to solve. Often, 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 \u2014 this is called combining like-terms.<\/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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>The 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 all the techniques you&#8217;ve learned.<\/p>\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]\\Large\\frac{12x}{\\color{red}{12}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141884&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\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-1\" 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<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 Answer<\/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]\\Large\\frac{-9}{\\color{red}{3}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.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<p>Let&#8217;s review how the Distributive Property works.<\/p>\n<div class=\"textbox shaded\">\n<h3>Distributive Property<\/h3>\n<p style=\"padding-left: 30px;\">If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 30px;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]3\\left(x+4\\right)[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/><\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot x+3\\cdot 4[\/latex]<\/p>\n<p>Now you give it a try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146473\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146473&theme=oea&iframe_resize_id=ohm146473&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Apply your understanding of the Distributive Property to an equation that needs to be solved.<\/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 Answer<\/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]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.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-2\" 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<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 Answer<\/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><\/p>\n<p><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 Answer<\/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]\\Large\\frac{8x}{\\color{red}{8}}\\normalsize =\\Large\\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>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:\/\/ohm.lumenlearning.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:\/\/ohm.lumenlearning.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 Answer<\/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]\\Large\\frac{-8}{\\color{red}{2}}\\normalsize =\\Large\\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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142125&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\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 Answer<\/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<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" style=\"height: 195px;\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]7x+5=6x+2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Collect the variable terms to the left side by subtracting [latex]6x[\/latex] from both sides.<\/td>\n<td style=\"height: 30px;\">[latex]7x\\color{red}{-6x}+5=6x\\color{red}{-6x}+2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]x+5=2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Now, collect the constants to the right side by subtracting [latex]5[\/latex] from both sides.<\/td>\n<td style=\"height: 30px;\">[latex]x+5\\color{red}{-5}=2\\color{red}{-5}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]x=-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">The solution is [latex]x=-3[\/latex] .<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Check:<\/td>\n<td style=\"height: 15px;\">[latex]7x+5=6x+2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Let [latex]x=-3[\/latex].<\/td>\n<td style=\"height: 30px;\">[latex]7(\\color{red}{-3})+5\\stackrel{\\text{?}}{=}6(\\color{red}{-3})+2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]-21+5\\stackrel{\\text{?}}{=}-18+2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]16=16\\quad\\checkmark[\/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=\"q399033\">Show Answer<\/span><\/p>\n<div id=\"q399033\" 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><\/td>\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]\\Large\\frac{9n}{\\color{red}{9}}\\normalsize =\\Large\\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-3\" 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 Answer<\/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<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><\/td>\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 [latex]1[\/latex] the coefficient of [latex]a[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{-15}{\\color{red}{3}}\\normalsize =\\Large\\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>Let [latex]a=-5[\/latex]<\/td>\n<td>[latex]2(\\color{red}{-5})-7\\stackrel{\\text{?}}{=}5(\\color{red}{-5})+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-4\" 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:\/\/ohm.lumenlearning.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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142136&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>Here is a much more complex multi-step equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]\\Large{\\frac{x+22}{3}}=2x+4[\/latex]<\/p>\n<table id=\"eip-id1168468709344\" class=\"unnumbered unstyled\" style=\"height: 195px;\" summary=\"The first line says 7x plus 5 equals 6x plus 2. The next line says\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]\\frac{x+22}{3}=2x+4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Multiply both sides of the equation by 3<\/td>\n<td style=\"height: 30px;\">[latex]\\color{red}{3}\\cdot\\frac{x+22}{3}=(2x+4)\\color{red}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Eliminate the [latex]\\frac{3}{3} = 1[\/latex] on the left and Distribute on the right<\/td>\n<td style=\"height: 15px;\">[latex]x+22=\\color{red}{3}(2x)+\\color{red}{3}(4)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Multiply<\/td>\n<td style=\"height: 30px;\">[latex]x+22=6x+12[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Subtract [latex]x[\/latex] from both sides (keeps variable term positive)<\/td>\n<td style=\"height: 30px;\">[latex]22=5x+12[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Subtract [latex]12[\/latex] from both sides (isolates the variable term)<\/td>\n<td style=\"height: 30px;\">[latex]10=5x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Divide both sides by [latex]5[\/latex] (isolates the variable)<\/td>\n<td style=\"height: 30px;\">[latex]2=x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Rewrite solution with variable on the right<\/td>\n<td style=\"height: 30px;\">[latex]x=2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">The solution is [latex]x=2[\/latex]<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Check:<\/td>\n<td style=\"height: 15px;\">[latex]\\frac{x+22}{3}=2x+4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Let [latex]x=-11[\/latex].<\/td>\n<td style=\"height: 30px;\">[latex]\\frac{\\color{red}{2}+22}{3}=2(\\color{red}{2})+4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]\\frac{24}{3}\\stackrel{\\text{?}}{=}4+4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]8=8\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/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\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-369\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"8326a2db-5c35-4add-bcd8-f2f6fc41c836, 10d3da92-8ce0-4c85-8fd5-2a28dfb9a6b1","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-369","chapter","type-chapter","status-publish","hentry"],"part":26,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/369","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/369\/revisions"}],"predecessor-version":[{"id":3513,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/369\/revisions\/3513"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/26"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/369\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=369"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=369"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=369"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}