{"id":3527,"date":"2023-05-25T14:15:37","date_gmt":"2023-05-25T14:15:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3527"},"modified":"2024-10-18T20:57:13","modified_gmt":"2024-10-18T20:57:13","slug":"modeling-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-background-youll-need-2\/","title":{"raw":"Introduction to Modeling: Background You'll Need 2","rendered":"Introduction to Modeling: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Solve equations by subtracting<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solve Equations Using the Subtraction Property of Equality<\/h2>\r\n<p>Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equation. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>subtraction property of equality<\/h3>\r\n<p>For any numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex]<\/p>\r\n<\/section>\r\n<p>Think about twin brothers Andy and Bobby. They are [latex]17[\/latex] years old. How old was Andy [latex]3[\/latex] years ago? He was [latex]3[\/latex] years less than [latex]17[\/latex], so his age was [latex]17 - 3[\/latex], or [latex]14[\/latex]. What about Bobby\u2019s age [latex]3[\/latex] years ago? Of course, he was [latex]14[\/latex] also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages [latex]3[\/latex] years ago.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a=b\\\\ a - 3=b - 3\\end{array}[\/latex]<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><b>How To: Solve an Equation Using the Subtraction Property of Equality<\/b><\/p>\r\n<ol id=\"eip-id1168469785635\" class=\"stepwise\">\r\n\t<li>Use the Subtraction Property of Equality to isolate the variable.<\/li>\r\n\t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n\t<li>Check the solution.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Solve: [latex]x+8=17[\/latex].[reveal-answer q=\"981126\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"981126\"]We will use the Subtraction Property of Equality to isolate [latex]x[\/latex].\r\n\r\n<table id=\"eip-id1168467165114\" class=\"unnumbered unstyled\" summary=\"The image shows the equation, x plus 8 equal to 17. Take 8 away from both sides of the equation to get x plus 8 minus 8 equal to 17 minus 8. On the left, plus 8 and minus 8 cancel out to leave just x. On the right 17 minus 8 is 9. The equation becomes x equal to 9.Back to the original equation, x plus 8 equal to 17. Substitute 9 in for x to check. The equation becomes 9 plus 8 equal to 17. Is this true? The left side simplifies by adding 9 and 8 to get 17. Both sides of the equal symbol are 17.\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">\u00a0<\/td>\r\n<td style=\"height: 15px;\">[latex]x+8=17[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Subtract [latex]8[\/latex] from both sides.<\/td>\r\n<td style=\"height: 15px;\">[latex]x+8\\color{red}{-8}\\color{black}=17\\color{red}{-8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"height: 20px;\">Simplify.<\/td>\r\n<td style=\"height: 20px;\">[latex]x=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 21.71875px;\">\r\n<td style=\"height: 21.71875px;\">\u00a0<\/td>\r\n<td style=\"height: 21.71875px;\">[latex]x+8=17[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"height: 22px;\">\u00a0<\/td>\r\n<td style=\"height: 22px;\">[latex]\\color{red}{9}\\color{black}+8=17[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"height: 22px;\">\u00a0<\/td>\r\n<td style=\"height: 22px;\">[latex]17=17\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Since [latex]x=9[\/latex] makes [latex]x+8=17[\/latex] a true statement, we know [latex]9[\/latex] is the solution to the equation.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6946[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Solve: [latex]100=y+74[\/latex].[reveal-answer q=\"932620\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"932620\"]To solve an equation, we must always isolate the variable\u2014it doesn\u2019t matter which side it is on. To isolate [latex]y[\/latex], we will subtract [latex]74[\/latex] from both sides.\r\n\r\n<table id=\"eip-id1168467375696\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation, 100 equal to y plus 74. Take 74 away from both sides of the equation to get 100 minus 74 equal to y plus 74 minus 74. On the left 100 minus 74 is 26. On the right plus 74 and minus 74 cancel out to leave just y. The equation becomes 26 equal to y.\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]100=y+74[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]74[\/latex] from both sides.<\/td>\r\n<td>[latex]100\\color{red}{-74}\\color{black}=y+74\\color{red}{-74}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]26=y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Substitute [latex]26[\/latex] for [latex]y[\/latex] to check.<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]100=y+74[\/latex]<\/p>\r\n<p>[latex]100=\\color{red}{26}\\color{black}+74[\/latex]<\/p>\r\n<p>[latex]100=100\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Since [latex]y=26[\/latex] makes [latex]100=y+74[\/latex] a true statement, we have found the solution to this equation.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6947[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Solve equations by subtracting<\/li>\n<\/ul>\n<\/section>\n<h2>Solve Equations Using the Subtraction Property of Equality<\/h2>\n<p>Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equation. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>subtraction property of equality<\/h3>\n<p>For any numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex]<\/p>\n<\/section>\n<p>Think about twin brothers Andy and Bobby. They are [latex]17[\/latex] years old. How old was Andy [latex]3[\/latex] years ago? He was [latex]3[\/latex] years less than [latex]17[\/latex], so his age was [latex]17 - 3[\/latex], or [latex]14[\/latex]. What about Bobby\u2019s age [latex]3[\/latex] years ago? Of course, he was [latex]14[\/latex] also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages [latex]3[\/latex] years ago.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a=b\\\\ a - 3=b - 3\\end{array}[\/latex]<\/p>\n<section class=\"textbox questionHelp\">\n<p><b>How To: Solve an Equation Using the Subtraction Property of Equality<\/b><\/p>\n<ol id=\"eip-id1168469785635\" class=\"stepwise\">\n<li>Use the Subtraction Property of Equality to isolate the variable.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Check the solution.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Solve: [latex]x+8=17[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981126\">Show Answer<\/button><\/p>\n<div id=\"q981126\" class=\"hidden-answer\" style=\"display: none\">We will use the Subtraction Property of Equality to isolate [latex]x[\/latex].<\/p>\n<table id=\"eip-id1168467165114\" class=\"unnumbered unstyled\" summary=\"The image shows the equation, x plus 8 equal to 17. Take 8 away from both sides of the equation to get x plus 8 minus 8 equal to 17 minus 8. On the left, plus 8 and minus 8 cancel out to leave just x. On the right 17 minus 8 is 9. The equation becomes x equal to 9.Back to the original equation, x plus 8 equal to 17. Substitute 9 in for x to check. The equation becomes 9 plus 8 equal to 17. Is this true? The left side simplifies by adding 9 and 8 to get 17. Both sides of the equal symbol are 17.\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">\u00a0<\/td>\n<td style=\"height: 15px;\">[latex]x+8=17[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Subtract [latex]8[\/latex] from both sides.<\/td>\n<td style=\"height: 15px;\">[latex]x+8\\color{red}{-8}\\color{black}=17\\color{red}{-8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px;\">Simplify.<\/td>\n<td style=\"height: 20px;\">[latex]x=9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 21.71875px;\">\n<td style=\"height: 21.71875px;\">\u00a0<\/td>\n<td style=\"height: 21.71875px;\">[latex]x+8=17[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px;\">\u00a0<\/td>\n<td style=\"height: 22px;\">[latex]\\color{red}{9}\\color{black}+8=17[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px;\">\u00a0<\/td>\n<td style=\"height: 22px;\">[latex]17=17\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Since [latex]x=9[\/latex] makes [latex]x+8=17[\/latex] a true statement, we know [latex]9[\/latex] is the solution to the equation.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6946\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6946&theme=lumen&iframe_resize_id=ohm6946&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Solve: [latex]100=y+74[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q932620\">Show Answer<\/button><\/p>\n<div id=\"q932620\" class=\"hidden-answer\" style=\"display: none\">To solve an equation, we must always isolate the variable\u2014it doesn\u2019t matter which side it is on. To isolate [latex]y[\/latex], we will subtract [latex]74[\/latex] from both sides.<\/p>\n<table id=\"eip-id1168467375696\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation, 100 equal to y plus 74. Take 74 away from both sides of the equation to get 100 minus 74 equal to y plus 74 minus 74. On the left 100 minus 74 is 26. On the right plus 74 and minus 74 cancel out to leave just y. The equation becomes 26 equal to y.\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]100=y+74[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]74[\/latex] from both sides.<\/td>\n<td>[latex]100\\color{red}{-74}\\color{black}=y+74\\color{red}{-74}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]26=y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Substitute [latex]26[\/latex] for [latex]y[\/latex] to check.<\/p>\n<\/td>\n<td>\n[latex]100=y+74[\/latex]<br \/>\n[latex]100=\\color{red}{26}\\color{black}+74[\/latex]<br \/>\n[latex]100=100\\quad\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Since [latex]y=26[\/latex] makes [latex]100=y+74[\/latex] a true statement, we have found the solution to this equation.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6947\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6947&theme=lumen&iframe_resize_id=ohm6947&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":87,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3527"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3527\/revisions"}],"predecessor-version":[{"id":14774,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3527\/revisions\/14774"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/87"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3527\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3527"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3527"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3527"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}