{"id":3876,"date":"2020-02-09T21:50:56","date_gmt":"2020-02-09T21:50:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3876"},"modified":"2021-02-05T23:55:44","modified_gmt":"2021-02-05T23:55:44","slug":"using-the-subtraction-and-addition-properties-for-multi-step-equations","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-the-subtraction-and-addition-properties-for-multi-step-equations\/","title":{"raw":"Using the Subtraction and Addition Properties for Multi-Step Equations","rendered":"Using the Subtraction and Addition Properties for Multi-Step Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality<\/li><li>Check your solution to a linear equation to verify its accuracy<\/li><\/ul><\/div>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.\n\n<div class=\"textbox exercises\"><h3>Example<\/h3>Solve:\n\n[latex]3x - 7 - 2x - 4=1[\/latex]\n\nSolution:\nThe left side of the equation has an expression that we should simplify before trying to isolate the variable.\n\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, \"><tbody><tr><td><\/td><td>[latex]3x-7-2x-4=1[\/latex]<\/td><\/tr><tr><td>Rearrange the terms, using the Commutative Property of Addition.<\/td><td>[latex]3x-2x-7-4=1[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]x-11=1[\/latex]<\/td><\/tr><tr><td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td><td>[latex]x-11\\color{red}{+11}=1\\color{red}{+11}[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]x=12[\/latex]<\/td><\/tr><tr><td>Check.Substitute [latex]x=12[\/latex] into the original equation.\n[latex]3x-7-2x-4=1[\/latex]\n\n[latex]3(\\color{red}{12})-7-2(\\color{red}{12})-4=1[\/latex]\n\n[latex]36-7-24-4=1[\/latex]\n\n[latex]29-24-4=1[\/latex]\n\n[latex]5-4=1[\/latex]\n\n[latex]1=1\\quad\\checkmark[\/latex]\n\nThe solution checks.\n\n<\/td><td><\/td><\/tr><\/tbody><\/table><\/div>Now you can try solving a couple&nbsp;equations where you should simplify first.\n\n<div class=\"textbox key-takeaways\"><h3>TRY&nbsp;IT<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\n\n\n\n<\/div>The 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.\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Solve: [latex]3\\left(n - 4\\right)-2n=-3[\/latex]\n[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"190834\"]\n\nSolution:\nThe left side of the equation has an expression that we should simplify.\n\n<table id=\"eip-id1168468254328\" class=\"unnumbered unstyled\" summary=\"The top line says \"><tbody><tr><td><\/td><td>[latex]3(n-4)-2n=-3[\/latex]<\/td><\/tr><tr><td>Distribute on the left.<\/td><td>[latex]3n-12-2n=-3[\/latex]<\/td><\/tr><tr><td>Use the Commutative Property to rearrange terms.<\/td><td>[latex]3n-2n-12=-3[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]n-12=-3[\/latex]<\/td><\/tr><tr><td>Isolate <em>n<\/em> using the Addition Property of Equality.<\/td><td>[latex]n-12\\color{red}{+12}=-3\\color{red}{+12}[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]n=9[\/latex]<\/td><\/tr><tr><td>Check.Substitute [latex]n=9[\/latex] into the original equation.\n[latex]3(n-4)-2n=-3[\/latex]\n[latex]3(\\color{red}{9}-4)-2\\cdot\\color{red}{9}=-3[\/latex]\n[latex]3(5)-18=-3[\/latex]\n[latex]15-18=-3[\/latex]\n[latex]-3=-3\\quad\\checkmark[\/latex]\nThe solution checks.\n\n<\/td><td><\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div>Now you can try a few problems that involve distribution.\n\n<div class=\"textbox key-takeaways\"><h3>TRY&nbsp;IT<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141737&amp;theme=oea&amp;iframe_resize_id=mom22[\/embed]\n\n\n\n<\/div>The next example has expressions on both sides that need to be simplified.\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Solve: [latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex]\n\n<p class=\"p1\">[reveal-answer q=\"190976\"]Show Solution[\/reveal-answer]\n\n<p class=\"p1\">[hidden-answer a=\"190976\"]\n\nSolution:\nBoth sides of the equation have expressions that we should simplify before we isolate the variable.\n\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 \"><tbody><tr><td><\/td><td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td><\/tr><tr><td>Distribute on the left, subtract on the right.<\/td><td>[latex]6k-2-5k=-9[\/latex]<\/td><\/tr><tr><td>Use the Commutative Property of Addition.<\/td><td>[latex]6k-5k-2=-9[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]k-2=-9[\/latex]<\/td><\/tr><tr><td>Undo subtraction by using the Addition Property of Equality.<\/td><td>[latex]k-2\\color{red}{+2}=-9\\color{red}{+2}[\/latex]<\/td><\/tr><tr><td>Simplify.<\/td><td>[latex]k=-7[\/latex]<\/td><\/tr><tr><td>Check.Let [latex]k=-7[\/latex].\n[latex]2(3k-1)-5k=-2-7[\/latex]\n[latex]2(3(\\color{red}{-7}-1)-5(\\color{red}{-7})=-2-7[\/latex]\n[latex]2(-21-1)-5(-7)=-9[\/latex]\n[latex]2(-22)+35=-9[\/latex]\n[latex]-44+35=-9[\/latex]\n[latex]-9=-9\\quad\\checkmark[\/latex]\n\n<\/td><\/tr><\/tbody><\/table>The solution checks.\n\n<p class=\"p1\">[\/hidden-answer]\n\n<\/div>Now, you give it a try!\n\n<div class=\"textbox key-takeaways\"><h3>TRY&nbsp;IT<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141739&amp;theme=oea&amp;iframe_resize_id=mom220[\/embed]\n\n\n\n<\/div>In the following video we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.\n\nhttps:\/\/youtu.be\/shGKzDBA5kQ\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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<\/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 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.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.<\/p>\n<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now you can try solving a couple&nbsp;equations where you should simplify first.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY&nbsp;IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141735\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&#38;theme=oea&#38;iframe_resize_id=ohm141735&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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.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.<\/p>\n<\/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&nbsp;IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141737&#38;theme=oea&#38;iframe_resize_id=ohm141737&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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.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]<\/p>\n<\/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&nbsp;IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141739\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141739&#38;theme=oea&#38;iframe_resize_id=ohm141739&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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\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-3876\">\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>Solve Linear Equations in One Variable with Simplifying (One-Step Add\/Subtract). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/shGKzDBA5kQ\">https:\/\/youtu.be\/shGKzDBA5kQ<\/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>Question ID 141735, 141737, 141739. <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":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solve Linear Equations in One Variable with Simplifying (One-Step Add\/Subtract)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/shGKzDBA5kQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 141735, 141737, 141739\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3876","chapter","type-chapter","status-web-only","hentry"],"part":356,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3876","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\/3876\/revisions"}],"predecessor-version":[{"id":3877,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3876\/revisions\/3877"}],"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\/3876\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3876"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3876"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3876"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3876"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}