{"id":3525,"date":"2023-05-25T14:15:21","date_gmt":"2023-05-25T14:15:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3525"},"modified":"2024-10-18T20:57:13","modified_gmt":"2024-10-18T20:57:13","slug":"modeling-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-background-youll-need-1\/","title":{"raw":"Introduction to Modeling: Background You'll Need 1","rendered":"Introduction to Modeling: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify numbers that are solutions to an equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6400,&quot;11&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Identify the number that is the solution to an equation<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\r\n<p>Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>solution of an equation<\/h3>\r\n<p>A solution to an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The process of finding the solution to an equation is called solving the equation.<\/p>\r\n<\/section>\r\n<p>To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of [latex]x+2=7?[\/latex] If you said [latex]5[\/latex], you\u2019re right! We say [latex]5[\/latex] is a solution to the equation [latex]x+2=7[\/latex] because when we substitute [latex]5[\/latex] for [latex]x[\/latex] the resulting statement is true.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill x+2=7\\hfill \\\\ \\hfill 5+2\\stackrel{?}{=}7\\hfill \\\\ \\\\ \\hfill 7=7\\quad\\checkmark \\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Since [latex]5+2=7[\/latex] is a true statement, we know that [latex]5[\/latex] is indeed a solution to the equation.<\/p>\r\n<p style=\"text-align: left;\">The symbol [latex]\\stackrel{?}{=}[\/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[\/latex] or not-equal sign [latex]\\not=[\/latex].<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Determine Whether a Number is a Solution to an Equation<\/strong><\/p>\r\n<ol id=\"eip-id1168468428753\" class=\"stepwise\">\r\n\t<li>Substitute the number for the variable in the equation.<\/li>\r\n\t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n\t<li>Determine whether the resulting equation is true.<br \/>\r\n<ul id=\"eip-409\">\r\n\t<li>If it is true, the number is a solution.<\/li>\r\n\t<li>If it is not true, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Determine whether [latex]x=5[\/latex] is a solution of [latex]6x - 17=16[\/latex].[reveal-answer q=\"565951\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"565951\"]\r\n\r\n<table id=\"eip-id1168469790662\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 x minus 17 equal to 16. Substitute 5 for x and the equation becomes 6 times 5 minus 17 equal to 16. Is this true? Simplify the left side of the equation by multiplying 6 by 5 to get 30. The left side becomes 30 minus 17 which is 13. Thirteen is not equal to 16 on the right side of the equation.\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]6x-17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{5}[\/latex] for [latex]x[\/latex].<\/td>\r\n<td>[latex]6\\cdot\\color{red}{5}\\color{black}-17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30-17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]13\\not=16[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>So [latex]x=5[\/latex] is not a solution to the equation [latex]6x - 17=16[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6944[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Determine whether [latex]y=2[\/latex] is a solution of [latex]6y - 4=5y - 2[\/latex].[reveal-answer q=\"366376\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"366376\"]Here, the variable appears on both sides of the equation. We must substitute [latex]2[\/latex] for each [latex]y[\/latex].\r\n\r\n<table id=\"eip-id1168469647895\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 y minus 4 equal to 5 y minus 2. Substitute 2 for y on both sides of the equation. The equation becomes 6 times 2 minus 4 equal to 5 times 2 minus 2. Is this true? Simplify the left side of the equation by multiplying 6 by 2 to get 12. Then subtract 4 from 12 to get 8. Simplify the right side of the equation by multiplying 5 by 2 to get 10. Then subtract 2 from 10 to get eight. Both sides of the equation are 8.\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]6y-4=5y-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{2}[\/latex] for [latex]y[\/latex].<\/td>\r\n<td>[latex]6(\\color{red}{2}\\color{black})-4=5(\\color{red}{2}\\color{black})-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]12-4=10-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]8=8\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Since [latex]y=2[\/latex] results in a true equation, we know that [latex]2[\/latex] is a solution to the equation [latex]6y - 4=5y - 2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6945[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify numbers that are solutions to an equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6400,&quot;11&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Identify the number that is the solution to an equation<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\n<p>Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>solution of an equation<\/h3>\n<p>A solution to an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p>\n<p>&nbsp;<\/p>\n<p>The process of finding the solution to an equation is called solving the equation.<\/p>\n<\/section>\n<p>To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of [latex]x+2=7?[\/latex] If you said [latex]5[\/latex], you\u2019re right! We say [latex]5[\/latex] is a solution to the equation [latex]x+2=7[\/latex] because when we substitute [latex]5[\/latex] for [latex]x[\/latex] the resulting statement is true.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill x+2=7\\hfill \\\\ \\hfill 5+2\\stackrel{?}{=}7\\hfill \\\\ \\\\ \\hfill 7=7\\quad\\checkmark \\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Since [latex]5+2=7[\/latex] is a true statement, we know that [latex]5[\/latex] is indeed a solution to the equation.<\/p>\n<p style=\"text-align: left;\">The symbol [latex]\\stackrel{?}{=}[\/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[\/latex] or not-equal sign [latex]\\not=[\/latex].<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Determine Whether a Number is a Solution to an Equation<\/strong><\/p>\n<ol id=\"eip-id1168468428753\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"eip-409\">\n<li>If it is true, the number is a solution.<\/li>\n<li>If it is not true, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Determine whether [latex]x=5[\/latex] is a solution of [latex]6x - 17=16[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q565951\">Show Answer<\/button><\/p>\n<div id=\"q565951\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469790662\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 x minus 17 equal to 16. Substitute 5 for x and the equation becomes 6 times 5 minus 17 equal to 16. Is this true? Simplify the left side of the equation by multiplying 6 by 5 to get 30. The left side becomes 30 minus 17 which is 13. Thirteen is not equal to 16 on the right side of the equation.\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]6x-17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{5}[\/latex] for [latex]x[\/latex].<\/td>\n<td>[latex]6\\cdot\\color{red}{5}\\color{black}-17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30-17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]13\\not=16[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>So [latex]x=5[\/latex] is not a solution to the equation [latex]6x - 17=16[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6944\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6944&theme=lumen&iframe_resize_id=ohm6944&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Determine whether [latex]y=2[\/latex] is a solution of [latex]6y - 4=5y - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q366376\">Show Answer<\/button><\/p>\n<div id=\"q366376\" class=\"hidden-answer\" style=\"display: none\">Here, the variable appears on both sides of the equation. We must substitute [latex]2[\/latex] for each [latex]y[\/latex].<\/p>\n<table id=\"eip-id1168469647895\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 y minus 4 equal to 5 y minus 2. Substitute 2 for y on both sides of the equation. The equation becomes 6 times 2 minus 4 equal to 5 times 2 minus 2. Is this true? Simplify the left side of the equation by multiplying 6 by 2 to get 12. Then subtract 4 from 12 to get 8. Simplify the right side of the equation by multiplying 5 by 2 to get 10. Then subtract 2 from 10 to get eight. Both sides of the equation are 8.\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]6y-4=5y-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{2}[\/latex] for [latex]y[\/latex].<\/td>\n<td>[latex]6(\\color{red}{2}\\color{black})-4=5(\\color{red}{2}\\color{black})-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]12-4=10-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]8=8\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Since [latex]y=2[\/latex] results in a true equation, we know that [latex]2[\/latex] is a solution to the equation [latex]6y - 4=5y - 2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6945\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6945&theme=lumen&iframe_resize_id=ohm6945&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"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\/3525"}],"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":19,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3525\/revisions"}],"predecessor-version":[{"id":14773,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3525\/revisions\/14773"}],"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\/3525\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3525"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3525"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3525"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3525"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}