{"id":1102,"date":"2024-05-04T22:27:39","date_gmt":"2024-05-04T22:27:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1102"},"modified":"2025-01-16T21:10:44","modified_gmt":"2025-01-16T21:10:44","slug":"modeling-with-linear-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/modeling-with-linear-equations-fresh-take\/","title":{"raw":"Modeling with Linear Equations: Fresh Take","rendered":"Modeling with Linear Equations: Fresh Take"},"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\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Modeling Real-World Situations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify known and unknown quantities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Assign variables to represent unknowns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Translate verbal descriptions into mathematical expressions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components of Linear Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): represents rate of change or variable cost<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept ([latex]b[\/latex]): represents initial value or fixed cost<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]y = mx + b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Real-World Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Cost analysis (fixed vs. variable costs)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Revenue and profit modeling<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Comparison of different options or plans<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inventory management<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Time and distance problems<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Steps in Problem-Solving<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify known and unknown quantities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Assign variables to unknowns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Express relationships between variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpret the solution in context<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">There are two cell phone companies that offer different packages. Company A charges a monthly service fee of [latex]$34[\/latex] plus [latex]$0.04[\/latex]\/min talk-time. Company B charges a monthly service fee of [latex]$45[\/latex] plus [latex]$0.03[\/latex]\/min talk-time. Use [latex]x[\/latex] for your variable.a) Write an equation that models the monthly cost for company A.b) Write an equation that models the monthly cost for company B.C) If the average number of minutes used each month is [latex]1,162[\/latex], how much is the monthly cost for each company?[reveal-answer q=\"981057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"981057\"]a) [latex]A=0.04x+34[\/latex]b) [latex]B = 0.03x+45[\/latex]c) Company A's cost would be [latex]$80.48[\/latex], and Company B's cost would be [latex]$79.86[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A cash register contains only five dollar and ten dollar bills. It contains twice as many five's as ten's and the total amount of money in the cash register is [latex]560 [\/latex] dollars. How many ten's are in the cash register?[reveal-answer q=\"315017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"315017\"]There are [latex]28[\/latex] ten dollar bills in the cash register.[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video shows another example of using linear equations to model and compare two cell phone plans.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahgbbech-Q5hlC_VPKGM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Q5hlC_VPKGM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahgbbech-Q5hlC_VPKGM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844231&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahgbbech-Q5hlC_VPKGM&vembed=0&video_id=Q5hlC_VPKGM&video_target=tpm-plugin-ahgbbech-Q5hlC_VPKGM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Write+Linear+Equations+to+Model+and+Compare+Cell+Phone+Plans+with+Data+Usage_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite Linear Equations to Model and Compare Cell Phone Plans with Data Usage\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for another example of using a linear equation in one variable to solve an application.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbbcahec-vOA7SGQCpr8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vOA7SGQCpr8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbbcahec-vOA7SGQCpr8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844232&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbbcahec-vOA7SGQCpr8&vembed=0&video_id=vOA7SGQCpr8&video_target=tpm-plugin-dbbcahec-vOA7SGQCpr8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solve+a+Coin+Problem+Using+an+Equation+in+One+Variable_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve a Coin Problem Using an Equation in One Variable\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using a Formula to Solve a Real-World Application<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Application of Known Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the appropriate formula for the given problem<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute known values into the formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for unknown variables<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Perimeter of a rectangle: [latex]P = 2L + 2W[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Area of a rectangle: [latex]A = LW[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Volume of a rectangular solid: [latex]V = LWH[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Distance-Rate-Time: [latex]d = rt[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multi-Step Problem Solving:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Break down complex problems into manageable steps<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use multiple formulas or equations when necessary<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Express one unknown in terms of another when needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Problem-Solving Steps<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the given information and the unknown(s)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Select the appropriate formula(s)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute known values into the formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the equation for the unknown(s)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution and interpret in context<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider unit conversions if necessary<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Strategies for Solving Formula-Based Problems<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Visualization:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Draw diagrams or sketches to represent the problem<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Label known and unknown quantities<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Unit Consistency:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Ensure all units are consistent before calculation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convert units if needed (e.g., minutes to hours)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Algebraic Manipulation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rearrange formulas to isolate the unknown variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use properties of equality to solve equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set up multiple equations for problems with more than one unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use substitution or elimination methods to solve systems<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reasonableness Check:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Verify if the solution makes sense in the context of the problem<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use estimation to check if the result is in the right ballpark<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Common Pitfalls to Avoid<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Misidentifying the appropriate formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Forgetting to convert units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Neglecting to consider all given information<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Misinterpreting the meaning of variables in context<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not checking the solution for reasonableness<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">On Saturday morning, it took Jennifer [latex]3.6[\/latex] hours to drive to her mother\u2019s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer [latex]4[\/latex] hours to return home. Her speed was [latex]5[\/latex] mi\/h slower on Sunday than on Saturday. What was her speed on Sunday?\r\n[reveal-answer q=\"929719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929719\"][latex]45 \\frac{\\text{mi}}{\\text{h}}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the dimensions of a rectangle given that the perimeter is [latex]110[\/latex] cm. and the length is [latex]1[\/latex] cm. more than twice the width.\r\n[reveal-answer q=\"447289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"447289\"][latex]L = 37 [\/latex] cm, [latex]W = 18[\/latex] cm[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">A game room has a perimeter of [latex]70[\/latex] ft. The length is five more than twice the width. How many ft<sup>2<\/sup> of new carpeting should be ordered?\r\n[reveal-answer q=\"646790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"646790\"][latex]250[\/latex] ft<sup>2<\/sup>[\/hidden-answer]<\/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<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Modeling Real-World Situations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify known and unknown quantities<\/li>\n<li class=\"whitespace-normal break-words\">Assign variables to represent unknowns<\/li>\n<li class=\"whitespace-normal break-words\">Translate verbal descriptions into mathematical expressions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components of Linear Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): represents rate of change or variable cost<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept ([latex]b[\/latex]): represents initial value or fixed cost<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]y = mx + b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Real-World Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Cost analysis (fixed vs. variable costs)<\/li>\n<li class=\"whitespace-normal break-words\">Revenue and profit modeling<\/li>\n<li class=\"whitespace-normal break-words\">Comparison of different options or plans<\/li>\n<li class=\"whitespace-normal break-words\">Inventory management<\/li>\n<li class=\"whitespace-normal break-words\">Time and distance problems<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Steps in Problem-Solving<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify known and unknown quantities<\/li>\n<li class=\"whitespace-normal break-words\">Assign variables to unknowns<\/li>\n<li class=\"whitespace-normal break-words\">Express relationships between variables<\/li>\n<li class=\"whitespace-normal break-words\">Write the equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve the equation<\/li>\n<li class=\"whitespace-normal break-words\">Interpret the solution in context<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">There are two cell phone companies that offer different packages. Company A charges a monthly service fee of [latex]$34[\/latex] plus [latex]$0.04[\/latex]\/min talk-time. Company B charges a monthly service fee of [latex]$45[\/latex] plus [latex]$0.03[\/latex]\/min talk-time. Use [latex]x[\/latex] for your variable.a) Write an equation that models the monthly cost for company A.b) Write an equation that models the monthly cost for company B.C) If the average number of minutes used each month is [latex]1,162[\/latex], how much is the monthly cost for each company?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981057\">Show Solution<\/button><\/p>\n<div id=\"q981057\" class=\"hidden-answer\" style=\"display: none\">a) [latex]A=0.04x+34[\/latex]b) [latex]B = 0.03x+45[\/latex]c) Company A&#8217;s cost would be [latex]$80.48[\/latex], and Company B&#8217;s cost would be [latex]$79.86[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A cash register contains only five dollar and ten dollar bills. It contains twice as many five&#8217;s as ten&#8217;s and the total amount of money in the cash register is [latex]560[\/latex] dollars. How many ten&#8217;s are in the cash register?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q315017\">Show Solution<\/button><\/p>\n<div id=\"q315017\" class=\"hidden-answer\" style=\"display: none\">There are [latex]28[\/latex] ten dollar bills in the cash register.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video shows another example of using linear equations to model and compare two cell phone plans.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahgbbech-Q5hlC_VPKGM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Q5hlC_VPKGM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahgbbech-Q5hlC_VPKGM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844231&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ahgbbech-Q5hlC_VPKGM&#38;vembed=0&#38;video_id=Q5hlC_VPKGM&#38;video_target=tpm-plugin-ahgbbech-Q5hlC_VPKGM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Write+Linear+Equations+to+Model+and+Compare+Cell+Phone+Plans+with+Data+Usage_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWrite Linear Equations to Model and Compare Cell Phone Plans with Data Usage\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for another example of using a linear equation in one variable to solve an application.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbbcahec-vOA7SGQCpr8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vOA7SGQCpr8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbbcahec-vOA7SGQCpr8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844232&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dbbcahec-vOA7SGQCpr8&#38;vembed=0&#38;video_id=vOA7SGQCpr8&#38;video_target=tpm-plugin-dbbcahec-vOA7SGQCpr8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solve+a+Coin+Problem+Using+an+Equation+in+One+Variable_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolve a Coin Problem Using an Equation in One Variable\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using a Formula to Solve a Real-World Application<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Application of Known Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the appropriate formula for the given problem<\/li>\n<li class=\"whitespace-normal break-words\">Substitute known values into the formula<\/li>\n<li class=\"whitespace-normal break-words\">Solve for unknown variables<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Perimeter of a rectangle: [latex]P = 2L + 2W[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Area of a rectangle: [latex]A = LW[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Volume of a rectangular solid: [latex]V = LWH[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Distance-Rate-Time: [latex]d = rt[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multi-Step Problem Solving:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Break down complex problems into manageable steps<\/li>\n<li class=\"whitespace-normal break-words\">Use multiple formulas or equations when necessary<\/li>\n<li class=\"whitespace-normal break-words\">Express one unknown in terms of another when needed<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Problem-Solving Steps<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the given information and the unknown(s)<\/li>\n<li class=\"whitespace-normal break-words\">Select the appropriate formula(s)<\/li>\n<li class=\"whitespace-normal break-words\">Substitute known values into the formula<\/li>\n<li class=\"whitespace-normal break-words\">Solve the equation for the unknown(s)<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution and interpret in context<\/li>\n<li class=\"whitespace-normal break-words\">Consider unit conversions if necessary<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Strategies for Solving Formula-Based Problems<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Visualization:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Draw diagrams or sketches to represent the problem<\/li>\n<li class=\"whitespace-normal break-words\">Label known and unknown quantities<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Unit Consistency:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ensure all units are consistent before calculation<\/li>\n<li class=\"whitespace-normal break-words\">Convert units if needed (e.g., minutes to hours)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Algebraic Manipulation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rearrange formulas to isolate the unknown variable<\/li>\n<li class=\"whitespace-normal break-words\">Use properties of equality to solve equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set up multiple equations for problems with more than one unknown<\/li>\n<li class=\"whitespace-normal break-words\">Use substitution or elimination methods to solve systems<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reasonableness Check:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Verify if the solution makes sense in the context of the problem<\/li>\n<li class=\"whitespace-normal break-words\">Use estimation to check if the result is in the right ballpark<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Common Pitfalls to Avoid<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Misidentifying the appropriate formula<\/li>\n<li class=\"whitespace-normal break-words\">Forgetting to convert units<\/li>\n<li class=\"whitespace-normal break-words\">Neglecting to consider all given information<\/li>\n<li class=\"whitespace-normal break-words\">Misinterpreting the meaning of variables in context<\/li>\n<li class=\"whitespace-normal break-words\">Not checking the solution for reasonableness<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">On Saturday morning, it took Jennifer [latex]3.6[\/latex] hours to drive to her mother\u2019s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer [latex]4[\/latex] hours to return home. Her speed was [latex]5[\/latex] mi\/h slower on Sunday than on Saturday. What was her speed on Sunday?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q929719\">Show Solution<\/button><\/p>\n<div id=\"q929719\" class=\"hidden-answer\" style=\"display: none\">[latex]45 \\frac{\\text{mi}}{\\text{h}}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the dimensions of a rectangle given that the perimeter is [latex]110[\/latex] cm. and the length is [latex]1[\/latex] cm. more than twice the width.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q447289\">Show Solution<\/button><\/p>\n<div id=\"q447289\" class=\"hidden-answer\" style=\"display: none\">[latex]L = 37[\/latex] cm, [latex]W = 18[\/latex] cm<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A game room has a perimeter of [latex]70[\/latex] ft. The length is five more than twice the width. How many ft<sup>2<\/sup> of new carpeting should be ordered?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q646790\">Show Solution<\/button><\/p>\n<div id=\"q646790\" class=\"hidden-answer\" style=\"display: none\">[latex]250[\/latex] ft<sup>2<\/sup><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Write Linear Equations to Model and Compare Cell Phone Plans with Data Usage \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=Q5hlC_VPKGM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solve a Coin Problem Using an Equation in One Variable \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=vOA7SGQCpr8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Write Linear Equations to Model and Compare Cell Phone Plans with Data Usage ","author":"","organization":"Mathispower4u","url":"https:\/\/www.youtube.com\/watch?v=Q5hlC_VPKGM","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Solve a Coin Problem Using an Equation in One Variable ","author":"","organization":"Mathispower4u","url":"https:\/\/www.youtube.com\/watch?v=vOA7SGQCpr8","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844231&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahgbbech-Q5hlC_VPKGM&vembed=0&video_id=Q5hlC_VPKGM&video_target=tpm-plugin-ahgbbech-Q5hlC_VPKGM'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844232&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbbcahec-vOA7SGQCpr8&vembed=0&video_id=vOA7SGQCpr8&video_target=tpm-plugin-dbbcahec-vOA7SGQCpr8'><\/script>\n","media_targets":["tpm-plugin-ahgbbech-Q5hlC_VPKGM","tpm-plugin-dbbcahec-vOA7SGQCpr8"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1102"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1102\/revisions"}],"predecessor-version":[{"id":7204,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1102\/revisions\/7204"}],"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\/1102\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1102"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1102"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1102"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}