{"id":9138,"date":"2017-05-02T14:55:37","date_gmt":"2017-05-02T14:55:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9138"},"modified":"2025-02-18T20:11:23","modified_gmt":"2025-02-18T20:11:23","slug":"putting-it-together-solving-multi-step-linear-equations-part-ii","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/putting-it-together-solving-multi-step-linear-equations-part-ii\/","title":{"raw":"Putting It Together: Multi-Step Linear Equations","rendered":"Putting It Together: Multi-Step Linear Equations"},"content":{"raw":"Kristy, the college student we met earlier who was anxiously studying for her final exam, still isn't sure what grade she needs to earn to pass her math class, and the\u00a0final is tomorrow! Let's\u00a0put your new math skills to good use\u00a0to help her figure it out.\r\n\r\nHere's what we already know:\r\n<ul>\r\n \t<li>Kristy's\u00a0current grade in the class is [latex]82.3[\/latex]%.<\/li>\r\n \t<li>She wants to know what score she needs on the final to keep her B in the class ([latex]\\ge80\\%[\/latex]).<\/li>\r\n \t<li>She also wants to know what score she needs on the final to pass the class with a C ([latex]\\ge70\\%[\/latex]).<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_124\" align=\"aligncenter\" width=\"410\"]<img class=\"wp-image-124 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1881\/2017\/06\/18173809\/testing.jpg\" alt=\"Several students at their desks in a classroom.\" width=\"410\" height=\"277\" \/> Exam time is coming.[\/caption]\r\n\r\nEven though Kristy knows her current grade is [latex]82.3[\/latex]%, that isn't enough information to determine what grade she\u00a0needs on the final. She also needs to know how much the final exam can affect her grade. To figure this out, she consults her class syllabus, where she finds the following breakdown of points in the class:\r\n<table class=\"shaded aligncenter\" style=\"width: 300px;\">\r\n<tbody>\r\n<tr>\r\n<td>Homework<\/td>\r\n<td>150 Points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quizzes<\/td>\r\n<td>100 Points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Midterm<\/td>\r\n<td>100 Points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Final Exam<\/td>\r\n<td>200 Points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>TOTAL<\/td>\r\n<td>550 Points<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHow can we put this information together in a way that will be useful?\r\n\r\nFirst, we know that Kristy has earned points for everything but the final exam. If we add up the points for homework, quizzes, and the midterm, we know there are [latex]350[\/latex] points possible so far ([latex]150+100+100[\/latex]). If Kristy's grade is currently [latex]82.3[\/latex]%, this means she has\u00a0earned [latex]82.3[\/latex]% of the total [latex]350[\/latex] points. We can multiply to find the number of points she has earned so far:\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]350 \\times 0.823 = 288.05[\/latex] points<\/span><\/p>\r\nSo we know that Kristy has [latex]288.05[\/latex] points. The next question to answer is: how many total points does she\u00a0need to achieve a B or a C?\r\n<p style=\"text-align: center;\">Typically a B is a total score of [latex]80[\/latex]% or [latex].8[\/latex], and a C is [latex]70[\/latex]% or [latex].7[\/latex]<\/p>\r\nThere are a total of [latex]550[\/latex] points in the class. We can use this to figure out how many points Kristy would need to get a certain grade:\r\n<p style=\"text-align: center;\">To get a B, she needs [latex]550 \\times 0.8 = 440[\/latex] points.<\/p>\r\n<p style=\"text-align: center;\">To get a C, she needs [latex]550 \\times 0.7 = 385[\/latex] points.<\/p>\r\n<span style=\"color: #000000;\">Now we can write a linear equation to determine the number\u00a0of points she needs\u00a0on the final to earn\u00a0either a B or a C in the class.<\/span>\r\n<p style=\"text-align: center;\">current number of points + points needed on the final = total points needed<\/p>\r\nAgain, Kristy has [latex]288.05[\/latex] points. Let's call the number of points she\u00a0needs on the final [latex]x[\/latex]. We know that\u00a0for a B, she\u00a0needs [latex]440[\/latex] points total. We can set up the equation and solve for [latex]x[\/latex] to determine the number of points she\u00a0needs on the final to earn a B.\r\n<p style=\"text-align: center;\">[latex]288.05+x=440[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]288.05+x\\color{red}{-288.05}=440\\color{red}{-288.05}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=151.95[\/latex]<\/p>\r\nTo state\u00a0that grade as a percentage, we can\u00a0divide it by the total number of points possible on the final exam: [latex]200[\/latex]. That means Kristy\u00a0needs [latex]151.95\/200=75.975[\/latex]%. In other words, a C grade of [latex]76[\/latex]% on the final will be enough for her to\u00a0get a\u00a0B in the class.\r\n\r\nNow let's calculate how many points she\u00a0can earn on the final to pass the class with a C. We'll use the same equation, but this time, we'll use [latex]385[\/latex] for the total number of points.\r\n<p style=\"text-align: center;\">[latex]288.05 + x = 385[\/latex]<\/p>\r\nSolve again, and you'll find that [latex]x = 96.95[\/latex]. If we divide that by [latex]200[\/latex] to figure out the percentage, we learn that Kristy needs a score of [latex]48.475[\/latex]% on the final to get a C in the class. In other words, she can fail the final exam earning a [latex]49[\/latex]% and still pass the class.\r\n\r\nKristy can now take the final exam knowing exactly how well she needs to do to pass the class and keep her scholarship--and it turns out that she doesn't need to be as nervous as she thought!","rendered":"<p>Kristy, the college student we met earlier who was anxiously studying for her final exam, still isn&#8217;t sure what grade she needs to earn to pass her math class, and the\u00a0final is tomorrow! Let&#8217;s\u00a0put your new math skills to good use\u00a0to help her figure it out.<\/p>\n<p>Here&#8217;s what we already know:<\/p>\n<ul>\n<li>Kristy&#8217;s\u00a0current grade in the class is [latex]82.3[\/latex]%.<\/li>\n<li>She wants to know what score she needs on the final to keep her B in the class ([latex]\\ge80\\%[\/latex]).<\/li>\n<li>She also wants to know what score she needs on the final to pass the class with a C ([latex]\\ge70\\%[\/latex]).<\/li>\n<\/ul>\n<div id=\"attachment_124\" style=\"width: 420px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-124\" class=\"wp-image-124\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1881\/2017\/06\/18173809\/testing.jpg\" alt=\"Several students at their desks in a classroom.\" width=\"410\" height=\"277\" \/><\/p>\n<p id=\"caption-attachment-124\" class=\"wp-caption-text\">Exam time is coming.<\/p>\n<\/div>\n<p>Even though Kristy knows her current grade is [latex]82.3[\/latex]%, that isn&#8217;t enough information to determine what grade she\u00a0needs on the final. She also needs to know how much the final exam can affect her grade. To figure this out, she consults her class syllabus, where she finds the following breakdown of points in the class:<\/p>\n<table class=\"shaded aligncenter\" style=\"width: 300px;\">\n<tbody>\n<tr>\n<td>Homework<\/td>\n<td>150 Points<\/td>\n<\/tr>\n<tr>\n<td>Quizzes<\/td>\n<td>100 Points<\/td>\n<\/tr>\n<tr>\n<td>Midterm<\/td>\n<td>100 Points<\/td>\n<\/tr>\n<tr>\n<td>Final Exam<\/td>\n<td>200 Points<\/td>\n<\/tr>\n<tr>\n<td>TOTAL<\/td>\n<td>550 Points<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>How can we put this information together in a way that will be useful?<\/p>\n<p>First, we know that Kristy has earned points for everything but the final exam. If we add up the points for homework, quizzes, and the midterm, we know there are [latex]350[\/latex] points possible so far ([latex]150+100+100[\/latex]). If Kristy&#8217;s grade is currently [latex]82.3[\/latex]%, this means she has\u00a0earned [latex]82.3[\/latex]% of the total [latex]350[\/latex] points. We can multiply to find the number of points she has earned so far:<\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]350 \\times 0.823 = 288.05[\/latex] points<\/span><\/p>\n<p>So we know that Kristy has [latex]288.05[\/latex] points. The next question to answer is: how many total points does she\u00a0need to achieve a B or a C?<\/p>\n<p style=\"text-align: center;\">Typically a B is a total score of [latex]80[\/latex]% or [latex].8[\/latex], and a C is [latex]70[\/latex]% or [latex].7[\/latex]<\/p>\n<p>There are a total of [latex]550[\/latex] points in the class. We can use this to figure out how many points Kristy would need to get a certain grade:<\/p>\n<p style=\"text-align: center;\">To get a B, she needs [latex]550 \\times 0.8 = 440[\/latex] points.<\/p>\n<p style=\"text-align: center;\">To get a C, she needs [latex]550 \\times 0.7 = 385[\/latex] points.<\/p>\n<p><span style=\"color: #000000;\">Now we can write a linear equation to determine the number\u00a0of points she needs\u00a0on the final to earn\u00a0either a B or a C in the class.<\/span><\/p>\n<p style=\"text-align: center;\">current number of points + points needed on the final = total points needed<\/p>\n<p>Again, Kristy has [latex]288.05[\/latex] points. Let&#8217;s call the number of points she\u00a0needs on the final [latex]x[\/latex]. We know that\u00a0for a B, she\u00a0needs [latex]440[\/latex] points total. We can set up the equation and solve for [latex]x[\/latex] to determine the number of points she\u00a0needs on the final to earn a B.<\/p>\n<p style=\"text-align: center;\">[latex]288.05+x=440[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]288.05+x\\color{red}{-288.05}=440\\color{red}{-288.05}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=151.95[\/latex]<\/p>\n<p>To state\u00a0that grade as a percentage, we can\u00a0divide it by the total number of points possible on the final exam: [latex]200[\/latex]. That means Kristy\u00a0needs [latex]151.95\/200=75.975[\/latex]%. In other words, a C grade of [latex]76[\/latex]% on the final will be enough for her to\u00a0get a\u00a0B in the class.<\/p>\n<p>Now let&#8217;s calculate how many points she\u00a0can earn on the final to pass the class with a C. We&#8217;ll use the same equation, but this time, we&#8217;ll use [latex]385[\/latex] for the total number of points.<\/p>\n<p style=\"text-align: center;\">[latex]288.05 + x = 385[\/latex]<\/p>\n<p>Solve again, and you&#8217;ll find that [latex]x = 96.95[\/latex]. If we divide that by [latex]200[\/latex] to figure out the percentage, we learn that Kristy needs a score of [latex]48.475[\/latex]% on the final to get a C in the class. In other words, she can fail the final exam earning a [latex]49[\/latex]% and still pass the class.<\/p>\n<p>Kristy can now take the final exam knowing exactly how well she needs to do to pass the class and keep her scholarship&#8211;and it turns out that she doesn&#8217;t need to be as nervous as she thought!<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9138\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Putting It Together: Solving Multi-Step Linear Equations Part I. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Students in a classroom. <strong>Provided by<\/strong>: US Department of Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.flickr.com\/photos\/departmentofed\/9599973167\">https:\/\/www.flickr.com\/photos\/departmentofed\/9599973167<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Students in a classroom\",\"author\":\"\",\"organization\":\"US Department of Education\",\"url\":\"https:\/\/www.flickr.com\/photos\/departmentofed\/9599973167\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Putting It Together: Solving Multi-Step Linear Equations Part I\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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\"}]","CANDELA_OUTCOMES_GUID":"f77ff678-8391-42e5-aee5-8027d90fd754","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9138","chapter","type-chapter","status-publish","hentry"],"part":7476,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9138\/revisions"}],"predecessor-version":[{"id":16154,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9138\/revisions\/16154"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/parts\/7476"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=9138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=9138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=9138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}