{"id":3529,"date":"2023-05-25T14:15:51","date_gmt":"2023-05-25T14:15:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3529"},"modified":"2024-10-18T20:57:14","modified_gmt":"2024-10-18T20:57:14","slug":"modeling-background-youll-need-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-background-youll-need-3\/","title":{"raw":"Introduction to Modeling: Background You'll Need 3","rendered":"Introduction to Modeling: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Solve equations by adding<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solve Equations Using the Addition Property of Equality<\/h2>\r\n<p>In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to \"undo\" the addition in order to isolate the variable.<\/p>\r\n<p>But suppose we have an equation with a number subtracted from the variable, such as [latex]x - 5=8[\/latex]. We want to isolate the variable, so to \"undo\" the subtraction we will add the number to both sides.<\/p>\r\n<p>We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>addition property of equality<\/h3>\r\n<p>For any numbers [latex]a,b[\/latex] , and [latex]c[\/latex],\u00a0if [latex]a=b[\/latex],\u00a0then [latex]a+c=b+c[\/latex]<\/p>\r\n<\/section>\r\n<p>Remember the [latex]17\\text{-year-old}[\/latex] twins, Andy and Bobby? In ten years, Andy\u2019s age will still equal Bobby\u2019s age. They will both be [latex]27[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a=b\\\\ a+10=b+10\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can add the same number to both sides and still keep the equality.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><b>How To: Solve an equation using the Addition Property of Equality<\/b><\/p>\r\n<ol id=\"eip-id1168469683666\" class=\"stepwise\">\r\n\t<li>Use the Addition 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 - 5=8[\/latex].<br \/>\r\n[reveal-answer q=\"835535\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"835535\"]We will use the Addition Property of Equality to isolate the variable.\r\n\r\n<table id=\"eip-id1168467172615\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"The image shows the given equation, x minus 5 equal to 8. Add 5 to both sides of the equation to get x minus 5 plus 5 equal to 8 plus 5. On the left, minus 5 plus 5 cancel out to leave just x. On the right 8 plus 5 is 13. The equation becomes x equal to 13. Back to the original equation, x minus 5 equal to 8. Substitute 13 in for x to check. The equation becomes 13 minus 5 equal to 8. Is this true? The left side simplifies by subtracting 5 from 13 to get 8. Both sides of the equal symbol are 8.\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 402px;\">\u00a0<\/td>\r\n<td style=\"width: 457px;\">[latex]x-5=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 402px;\">Add [latex]5[\/latex] to both sides.<\/td>\r\n<td style=\"width: 457px;\">[latex]x-5\\color{red}{+5}\\color{black}=8\\color{red}{+5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 402px;\">Simplify.<\/td>\r\n<td style=\"width: 457px;\">[latex]x=13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 402px;\">Now we can check. Let [latex]x=\\color{red}{13}[\/latex].<\/td>\r\n<td style=\"width: 457px;\">[latex]x-5=8[\/latex] [latex]\\color{red}{13}\\color{black}-5=8[\/latex] [latex]8=8\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6948[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Solve: [latex]27=a - 16[\/latex].<br \/>\r\n[reveal-answer q=\"917467\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"917467\"]We will add [latex]16[\/latex] to each side to isolate the variable.\r\n\r\n<table id=\"eip-id1168468296088\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation, 27 equal to a minus 16. Add 16 to both sides of the equation to get 27 plus 16 equal to a minus 16 plus 16. On the left 27 plus 16 is 43. On the right, minus 16 plus 16 cancel out to leave just a. The equation becomes 43 equal to a. Back to the original equation,27 equal to a minus 16. Substitute 43 in for a to check. The equation becomes 27 equal to 43 minus 16. Is this true? The right side simplifies by subtracting 16 from 43 to get 27. Both sides of the equal symbol are 27.\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]27=a-16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]16[\/latex] to each side.<\/td>\r\n<td>[latex]27\\color{red}{+16}\\color{black}=a-16\\color{red}{+16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]43=a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now we can check. Let [latex]a=\\color{red}{43}[\/latex].<\/td>\r\n<td>[latex]27=a-16[\/latex] [latex]27=\\color{red}{43}\\color{black}-16[\/latex] [latex]27=27\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>The solution to [latex]27=a - 16[\/latex] is [latex]a=43[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6949[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Solve equations by adding<\/li>\n<\/ul>\n<\/section>\n<h2>Solve Equations Using the Addition Property of Equality<\/h2>\n<p>In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to &#8220;undo&#8221; the addition in order to isolate the variable.<\/p>\n<p>But suppose we have an equation with a number subtracted from the variable, such as [latex]x - 5=8[\/latex]. We want to isolate the variable, so to &#8220;undo&#8221; the subtraction we will add the number to both sides.<\/p>\n<p>We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>addition property of equality<\/h3>\n<p>For any numbers [latex]a,b[\/latex] , and [latex]c[\/latex],\u00a0if [latex]a=b[\/latex],\u00a0then [latex]a+c=b+c[\/latex]<\/p>\n<\/section>\n<p>Remember the [latex]17\\text{-year-old}[\/latex] twins, Andy and Bobby? In ten years, Andy\u2019s age will still equal Bobby\u2019s age. They will both be [latex]27[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a=b\\\\ a+10=b+10\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">We can add the same number to both sides and still keep the equality.<\/p>\n<section class=\"textbox questionHelp\">\n<p><b>How To: Solve an equation using the Addition Property of Equality<\/b><\/p>\n<ol id=\"eip-id1168469683666\" class=\"stepwise\">\n<li>Use the Addition 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 - 5=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q835535\">Show Answer<\/button><\/p>\n<div id=\"q835535\" class=\"hidden-answer\" style=\"display: none\">We will use the Addition Property of Equality to isolate the variable.<\/p>\n<table id=\"eip-id1168467172615\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"The image shows the given equation, x minus 5 equal to 8. Add 5 to both sides of the equation to get x minus 5 plus 5 equal to 8 plus 5. On the left, minus 5 plus 5 cancel out to leave just x. On the right 8 plus 5 is 13. The equation becomes x equal to 13. Back to the original equation, x minus 5 equal to 8. Substitute 13 in for x to check. The equation becomes 13 minus 5 equal to 8. Is this true? The left side simplifies by subtracting 5 from 13 to get 8. Both sides of the equal symbol are 8.\">\n<tbody>\n<tr>\n<td style=\"width: 402px;\">\u00a0<\/td>\n<td style=\"width: 457px;\">[latex]x-5=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 402px;\">Add [latex]5[\/latex] to both sides.<\/td>\n<td style=\"width: 457px;\">[latex]x-5\\color{red}{+5}\\color{black}=8\\color{red}{+5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 402px;\">Simplify.<\/td>\n<td style=\"width: 457px;\">[latex]x=13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 402px;\">Now we can check. Let [latex]x=\\color{red}{13}[\/latex].<\/td>\n<td style=\"width: 457px;\">[latex]x-5=8[\/latex] [latex]\\color{red}{13}\\color{black}-5=8[\/latex] [latex]8=8\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6948\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6948&theme=lumen&iframe_resize_id=ohm6948&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Solve: [latex]27=a - 16[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q917467\">Show Answer<\/button><\/p>\n<div id=\"q917467\" class=\"hidden-answer\" style=\"display: none\">We will add [latex]16[\/latex] to each side to isolate the variable.<\/p>\n<table id=\"eip-id1168468296088\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation, 27 equal to a minus 16. Add 16 to both sides of the equation to get 27 plus 16 equal to a minus 16 plus 16. On the left 27 plus 16 is 43. On the right, minus 16 plus 16 cancel out to leave just a. The equation becomes 43 equal to a. Back to the original equation,27 equal to a minus 16. Substitute 43 in for a to check. The equation becomes 27 equal to 43 minus 16. Is this true? The right side simplifies by subtracting 16 from 43 to get 27. Both sides of the equal symbol are 27.\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]27=a-16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]16[\/latex] to each side.<\/td>\n<td>[latex]27\\color{red}{+16}\\color{black}=a-16\\color{red}{+16}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]43=a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Now we can check. Let [latex]a=\\color{red}{43}[\/latex].<\/td>\n<td>[latex]27=a-16[\/latex] [latex]27=\\color{red}{43}\\color{black}-16[\/latex] [latex]27=27\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The solution to [latex]27=a - 16[\/latex] is [latex]a=43[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6949\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6949&theme=lumen&iframe_resize_id=ohm6949&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"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\"}]","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":[{"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"}],"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\/3529"}],"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":16,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3529\/revisions"}],"predecessor-version":[{"id":14775,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3529\/revisions\/14775"}],"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\/3529\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3529"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3529"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3529"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}