{"id":3878,"date":"2020-02-09T21:57:37","date_gmt":"2020-02-09T21:57:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3878"},"modified":"2021-02-05T23:55:46","modified_gmt":"2021-02-05T23:55:46","slug":"using-the-division-and-multiplication-properties-of-equality-for-multi-step-equations","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-the-division-and-multiplication-properties-of-equality-for-multi-step-equations\/","title":{"raw":"Using the Division and Multiplication Properties of Equality for Multi-Step Equations","rendered":"Using the Division and Multiplication Properties of Equality for Multi-Step Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Solve a linear equation that requires simplification before using properties of equality<\/li><li>Solve a linear equation that requires a combination of the properties of equality<\/li><\/ul><\/div>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's work through an example that will employ the following techniques:\n\n<ul><li>simplify by combining like terms<\/li><li>isolate x by using the division property of equality<\/li><\/ul><div class=\"textbox exercises\"><h3>Example<\/h3>Solve: [latex]8x+9x - 5x=-3+15[\/latex]\n\nSolution:\n\nFirst, we need to simplify both sides of the equation as much as possible\n\nStart by combining like terms to simplify each side.\n\n<table id=\"eip-id1168466098204\" class=\"unnumbered unstyled\" summary=\"The first line says \"><tbody><tr><td><\/td><td>[latex]8x+9x-5x=-3+15[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]12x=12[\/latex]<\/td><\/tr><tr><td>Divide both sides by 12 to isolate x.<\/td><td>[latex]\\Large\\frac{12x}{\\color{red}{12}}\\normalsize =\\Large\\frac{12}{\\color{red}{12}}[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]x=1[\/latex]<\/td><\/tr><tr><td>Check your answer. Let [latex]x=1[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]8x+9x-5x=-3+15[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]8\\cdot\\color{red}{1}+9\\cdot\\color{red}{1}-5\\cdot\\color{red}{1}\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]8+9-5\\stackrel{\\text{?}}{=}-3+15[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]12=12\\quad\\checkmark[\/latex]<\/td><td><\/td><\/tr><\/tbody><\/table><\/div>Here is a similar problem for you to try.\n\n<div class=\"textbox key-takeaways\"><h3>Try&nbsp;it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141884&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/div>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.\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Solve: [latex]11 - 20=17y - 8y - 6y[\/latex]\n\n[reveal-answer q=\"399032\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"399032\"]\n\nSolution:\n\nSimplify each side by combining like terms.\n\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 \"><tbody><tr><td><\/td><td>[latex]11-20=17y-8y-6y[\/latex]<\/td><\/tr><tr><td>Simplify each side.<\/td><td>[latex]-9=3y[\/latex]<\/td><\/tr><tr><td>Divide both sides by 3 to isolate y.<\/td><td>[latex]\\Large\\frac{-9}{\\color{red}{3}}\\normalsize =\\Large\\frac{3y}{\\color{red}{3}}[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]-3=y[\/latex]<\/td><\/tr><tr><td>Check your answer. Let [latex]y=-3[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]11-20=17y-8y-6y[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]11-20\\stackrel{\\text{?}}{=}17(\n\\color{red}{-3})-8(\\color{red}{-3})-6(\\color{red}{-3})[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]11-20\\stackrel{\\text{?}}{=}-51+24+18[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]-9=-9\\quad\\checkmark[\/latex]<\/td><td><\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div>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.\n\nNow you can try solving a similar problem.\n\n<div class=\"textbox key-takeaways\"><h3>try&nbsp;it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141901&amp;theme=oea&amp;iframe_resize_id=mom23[\/embed]\n\n\n\n<\/div>In our next example, we have an equation that contains a set of parentheses. &nbsp;We will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Solve: [latex]-3\\left(n - 2\\right)-6=21[\/latex]\n\nRemember\u2014always simplify each side first.\n[reveal-answer q=\"789987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"789987\"]\n\nSolution:\n\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 \"><tbody><tr><td><\/td><td>[latex]-3(n-2)-6=21[\/latex]<\/td><\/tr><tr><td>Distribute.<\/td><td>[latex]-3n+6-6=21[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]-3n=21[\/latex]<\/td><\/tr><tr><td>Divide both sides by -3 to isolate n.<\/td><td>[latex]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\frac{21}{\\color{red}{-3}}[\/latex][latex]n=-7[\/latex]\n\n<\/td><\/tr><tr><td>Check your answer. Let [latex]n=-7[\/latex] .<\/td><td><\/td><\/tr><tr><td>[latex]-3(n-2)-6=21[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]-3(\\color{red}{-7}-2)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]-3(-9)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]27-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td><td><\/td><\/tr><tr><td>[latex]21=21\\quad\\checkmark[\/latex]<\/td><td><\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div>Now you can try a similar problem.\n\n<div class=\"textbox key-takeaways\"><h3>try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141911&amp;theme=oea&amp;iframe_resize_id=mom27[\/embed]\n\n\n\n<\/div>In the following&nbsp;video you will see another example of using the division property of equality to solve an equation as well as &nbsp;another example of how to solve a multi-step equation that includes a set of parentheses.\n\nhttps:\/\/youtu.be\/qe89pkRKzRw\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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<\/ul>\n<\/div>\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]\\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&nbsp;it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141884\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141884&#38;theme=oea&#38;iframe_resize_id=ohm141884&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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]\\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&nbsp;it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141901\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141901&#38;theme=oea&#38;iframe_resize_id=ohm141901&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In our next example, we have an equation that contains a set of parentheses. &nbsp;We 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]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\frac{21}{\\color{red}{-3}}[\/latex][latex]n=-7[\/latex]<\/p>\n<\/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=\"ohm141911\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141911&#38;theme=oea&#38;iframe_resize_id=ohm141911&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following&nbsp;video you will see another example of using the division property of equality to solve an equation as well as &nbsp;another example of how to solve a multi-step equation that includes a set of parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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\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-3878\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Solving an Equation with One Set of Parentheses. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aQOkD8L57V0\">https:\/\/youtu.be\/aQOkD8L57V0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solve Linear Equations in One Variable with Simplifying (One-Step Mult\/Div). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qe89pkRKzRw\">https:\/\/youtu.be\/qe89pkRKzRw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 141884, 141901, 141911. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solving an Equation with One Set of Parentheses\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3878","chapter","type-chapter","status-web-only","hentry"],"part":356,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3878","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3878\/revisions"}],"predecessor-version":[{"id":3879,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3878\/revisions\/3879"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/356"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3878\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3878"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3878"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3878"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3878"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}