{"id":1069,"date":"2024-05-04T00:50:34","date_gmt":"2024-05-04T00:50:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1069"},"modified":"2024-12-03T17:00:03","modified_gmt":"2024-12-03T17:00:03","slug":"modeling-with-linear-equations-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/modeling-with-linear-equations-learn-it-1\/","title":{"raw":"Modeling with Linear Equations: Learn It 1","rendered":"Modeling with Linear Equations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4995,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16776960},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Setting up a Linear Equation to Solve a Real-World Application<\/h2>\r\nMany real-world applications can be modeled by linear equations. For example: Josh is hoping to get an A in his college algebra class. He has scores of [latex]75, 82, 95, 91,[\/latex] and [latex]94[\/latex] on his first five tests. Only the final exam remains, and the maximum of points that can be earned is [latex]100[\/latex]. Is it possible for Josh to end the course with an A? A linear equation will give Josh his answer.\r\n\r\nTo set up a linear equation that models a real-world situation, we must first determine the known quantities and define the unknown quantity as a variable. Then, we interpret the words as mathematical expressions using mathematical symbols. If a quantity is independent of a variable, we usually just add or subtract it according to the problem.\r\n\r\nWhen dealing with real-world applications, there are certain expressions that we can translate directly into math. The table below lists some common verbal expressions and their equivalent mathematical expressions.\r\n<table summary=\"A table with 8 rows and 2 columns. The entries in the first row are: Verbal and Translation to math operations. The entries in the second row are: One number exceeds another by a and x, x+a. The entries in the third row are: Twice a number and 2x. The entries in the fourth row are: One number is a more than another number and x, x plus a. The entries in the fifth row are: One number is a less than twice another number and x,2 times x minus a. The entries in the sixth row are: The product of a number and a, decreased by b and a times x minus b. The entries in the seventh row are: The quotient of a number and the number plus a is three times the number and x divided by the quantity x plus a equals three times x. The entries in the eighth row are: The product of three times a number and the number decreased by b is c and three times x times the quantity x minus b equals c.\">\r\n<thead>\r\n<tr>\r\n<th>Verbal<\/th>\r\n<th>Translation to Math Operations<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>One number exceeds another by <em>a<\/em><\/td>\r\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Twice a number<\/td>\r\n<td>[latex]2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>One number is <em>a <\/em>more than another number<\/td>\r\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>One number is <em>a <\/em>less than twice another number<\/td>\r\n<td>[latex]x,2x-a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of a number and <em>a<\/em>, decreased by <em>b<\/em><\/td>\r\n<td>[latex]ax-b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The quotient of a number and the number plus <em>a <\/em>is three times the number<\/td>\r\n<td>[latex]\\Large\\frac{x}{x+a}\\normalsize =3x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of three times a number and the number decreased by <em>b <\/em>is <em>c<\/em><\/td>\r\n<td>[latex]3x\\left(x-b\\right)=c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\" aria-label=\"Example\">Consider a car rental agency that charges [latex]$150[\/latex] per week plus [latex]$0.10[\/latex] per mile driven to rent a compact car. We can use these quantities to write an equation that models the cost of renting the car for a week [latex]C[\/latex] given a certain number of miles [latex]x[\/latex] driven.\r\n<div style=\"text-align: center;\">[latex]C=150+0.10x[\/latex]<\/div>\r\n<div>\r\n\r\nIn this case, a known cost, such as [latex]$0.10[\/latex] per mile, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[\/latex] to model the portion of the weekly cost generated by miles driven. This expression represents a variable cost because it changes according to the number of miles driven. There is a flat fee of [latex] $150[\/latex] to rent the car, independent of the number of miles driven. In applications involving costs, amounts such as this flat fee that do not change are often called fixed costs.\r\n\r\n<\/div>\r\n<\/section>In this example, we identified the unknown quantity as the number of miles driven and assigned it the variable [latex]x[\/latex]. Next, we identified the known quantities and translated the given information into an equation that models the total weekly cost. The equation [latex]C=150+0.10x[\/latex] shows the relationship between the fixed cost of [latex]$150[\/latex] to rent the car and the additional cost of [latex]$0.10[\/latex] per mile driven. With this model, we can answer questions like how much it would cost to drive [latex]500[\/latex] miles in a week or how many miles you could drive on a [latex]$375[\/latex] budget.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a real-world situation, write a linear equation to model it<\/strong>\r\n<ol>\r\n \t<li>Identify known and unknown quantities.<\/li>\r\n \t<li>Assign a variable to represent the unknown quantity.<\/li>\r\n \t<li>If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.<\/li>\r\n \t<li>Write an equation interpreting the words in the problem as mathematical operations.<\/li>\r\n \t<li>Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Find a linear equation to solve for the following unknown quantities:\r\n<center>One number exceeds another number by [latex]17[\/latex] and their sum is [latex]31[\/latex].<\/center>\r\nThen, find the two unknown values.\r\n\r\n<hr \/>\r\n\r\n<strong>Step 1: Define the variable.<\/strong>\r\n<ul>\r\n \t<li>Let [latex]x[\/latex] equal the first number.<\/li>\r\n \t<li>Since the second number exceeds the first by [latex]17[\/latex], we write the second number as [latex]x+17[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Step 2: Set up the equation.<\/strong>\r\n\r\nThe sum of the two numbers is [latex]31[\/latex], which leads us to the equation:\r\n\r\n<center>[latex]\\begin{align*} \\text{First number} + \\text{Second number} &amp;= 31 \\\\ x + (x + 17) &amp;= 31 \\end{align*}[\/latex]<\/center><strong>Step 3: Simplify and solve the equation.<\/strong>\r\n\r\n<center>[latex]\\begin{align*} x + (x + 17) &amp;= 31 &amp; \\text{Add the first and second number} \\\\ 2x + 17 &amp;= 31 &amp; \\text{Combine like terms} \\\\ 2x &amp;= 14 &amp; \\text{Subtract 17 from both sides} \\\\ x &amp;= 7 &amp; \\text{Divide both sides by 2} \\end{align*}[\/latex]<\/center>\r\n<ul>\r\n \t<li>The first number is [latex]x=7[\/latex].<\/li>\r\n \t<li>The second number would then be [latex]x+17=7+17=24[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18935[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18936[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4995,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16776960},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Setting up a Linear Equation to Solve a Real-World Application<\/h2>\n<p>Many real-world applications can be modeled by linear equations. For example: Josh is hoping to get an A in his college algebra class. He has scores of [latex]75, 82, 95, 91,[\/latex] and [latex]94[\/latex] on his first five tests. Only the final exam remains, and the maximum of points that can be earned is [latex]100[\/latex]. Is it possible for Josh to end the course with an A? A linear equation will give Josh his answer.<\/p>\n<p>To set up a linear equation that models a real-world situation, we must first determine the known quantities and define the unknown quantity as a variable. Then, we interpret the words as mathematical expressions using mathematical symbols. If a quantity is independent of a variable, we usually just add or subtract it according to the problem.<\/p>\n<p>When dealing with real-world applications, there are certain expressions that we can translate directly into math. The table below lists some common verbal expressions and their equivalent mathematical expressions.<\/p>\n<table summary=\"A table with 8 rows and 2 columns. The entries in the first row are: Verbal and Translation to math operations. The entries in the second row are: One number exceeds another by a and x, x+a. The entries in the third row are: Twice a number and 2x. The entries in the fourth row are: One number is a more than another number and x, x plus a. The entries in the fifth row are: One number is a less than twice another number and x,2 times x minus a. The entries in the sixth row are: The product of a number and a, decreased by b and a times x minus b. The entries in the seventh row are: The quotient of a number and the number plus a is three times the number and x divided by the quantity x plus a equals three times x. The entries in the eighth row are: The product of three times a number and the number decreased by b is c and three times x times the quantity x minus b equals c.\">\n<thead>\n<tr>\n<th>Verbal<\/th>\n<th>Translation to Math Operations<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>One number exceeds another by <em>a<\/em><\/td>\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Twice a number<\/td>\n<td>[latex]2x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>One number is <em>a <\/em>more than another number<\/td>\n<td>[latex]x,\\text{ }x+a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>One number is <em>a <\/em>less than twice another number<\/td>\n<td>[latex]x,2x-a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of a number and <em>a<\/em>, decreased by <em>b<\/em><\/td>\n<td>[latex]ax-b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The quotient of a number and the number plus <em>a <\/em>is three times the number<\/td>\n<td>[latex]\\Large\\frac{x}{x+a}\\normalsize =3x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of three times a number and the number decreased by <em>b <\/em>is <em>c<\/em><\/td>\n<td>[latex]3x\\left(x-b\\right)=c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\" aria-label=\"Example\">Consider a car rental agency that charges [latex]$150[\/latex] per week plus [latex]$0.10[\/latex] per mile driven to rent a compact car. We can use these quantities to write an equation that models the cost of renting the car for a week [latex]C[\/latex] given a certain number of miles [latex]x[\/latex] driven.<\/p>\n<div style=\"text-align: center;\">[latex]C=150+0.10x[\/latex]<\/div>\n<div>\n<p>In this case, a known cost, such as [latex]$0.10[\/latex] per mile, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[\/latex] to model the portion of the weekly cost generated by miles driven. This expression represents a variable cost because it changes according to the number of miles driven. There is a flat fee of [latex]$150[\/latex] to rent the car, independent of the number of miles driven. In applications involving costs, amounts such as this flat fee that do not change are often called fixed costs.<\/p>\n<\/div>\n<\/section>\n<p>In this example, we identified the unknown quantity as the number of miles driven and assigned it the variable [latex]x[\/latex]. Next, we identified the known quantities and translated the given information into an equation that models the total weekly cost. The equation [latex]C=150+0.10x[\/latex] shows the relationship between the fixed cost of [latex]$150[\/latex] to rent the car and the additional cost of [latex]$0.10[\/latex] per mile driven. With this model, we can answer questions like how much it would cost to drive [latex]500[\/latex] miles in a week or how many miles you could drive on a [latex]$375[\/latex] budget.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a real-world situation, write a linear equation to model it<\/strong><\/p>\n<ol>\n<li>Identify known and unknown quantities.<\/li>\n<li>Assign a variable to represent the unknown quantity.<\/li>\n<li>If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.<\/li>\n<li>Write an equation interpreting the words in the problem as mathematical operations.<\/li>\n<li>Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Find a linear equation to solve for the following unknown quantities:<\/p>\n<div style=\"text-align: center;\">One number exceeds another number by [latex]17[\/latex] and their sum is [latex]31[\/latex].<\/div>\n<p>Then, find the two unknown values.<\/p>\n<hr \/>\n<p><strong>Step 1: Define the variable.<\/strong><\/p>\n<ul>\n<li>Let [latex]x[\/latex] equal the first number.<\/li>\n<li>Since the second number exceeds the first by [latex]17[\/latex], we write the second number as [latex]x+17[\/latex].<\/li>\n<\/ul>\n<p><strong>Step 2: Set up the equation.<\/strong><\/p>\n<p>The sum of the two numbers is [latex]31[\/latex], which leads us to the equation:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*} \\text{First number} + \\text{Second number} &= 31 \\\\ x + (x + 17) &= 31 \\end{align*}[\/latex]<\/div>\n<p><strong>Step 3: Simplify and solve the equation.<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*} x + (x + 17) &= 31 & \\text{Add the first and second number} \\\\ 2x + 17 &= 31 & \\text{Combine like terms} \\\\ 2x &= 14 & \\text{Subtract 17 from both sides} \\\\ x &= 7 & \\text{Divide both sides by 2} \\end{align*}[\/latex]<\/div>\n<ul>\n<li>The first number is [latex]x=7[\/latex].<\/li>\n<li>The second number would then be [latex]x+17=7+17=24[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18935\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18935&theme=lumen&iframe_resize_id=ohm18935&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18936\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18936&theme=lumen&iframe_resize_id=ohm18936&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"learn_it","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1069"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1069\/revisions"}],"predecessor-version":[{"id":6551,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1069\/revisions\/6551"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1069\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1069"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1069"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1069"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}