{"id":2436,"date":"2018-12-11T13:41:21","date_gmt":"2018-12-11T18:41:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-applications-with-systems-of-equations\/"},"modified":"2019-05-22T19:31:10","modified_gmt":"2019-05-22T23:31:10","slug":"solve-applications-with-systems-of-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-applications-with-systems-of-equations\/","title":{"raw":"Solve Applications with Systems of Equations","rendered":"Solve Applications with Systems of Equations"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Solve direct translation applications<\/li>\r\n \t<li>Solve geometry applications<\/li>\r\n \t<li>Solve uniform motion applications<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167830703077\" class=\"be-prepared\">\r\n<p id=\"fs-id1167830698775\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1167835341827\" type=\"1\">\r\n \t<li>The sum of twice a number and nine is 31. Find the number.\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766#fs-id1167836525150\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Twins Jon and Ron together earned ?96,000 last year. Ron earned ?8000 more than three times what Jon earned. How much did each of the twins earn?\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766#fs-id1167836606849\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains.\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2#fs-id1167836309166\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Solve Direct Translation Applications<\/h3>\r\n<p id=\"fs-id1167835302397\">Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we\u2019ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167828435685\" class=\"howto\">\r\n<div data-type=\"title\">Solve applications with systems of equations.<\/div>\r\n<ol id=\"fs-id1167834120308\" class=\"stepwise\" type=\"1\">\r\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\r\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\r\n \t<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\r\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\r\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\r\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1167830865965\">We solved number problems with one variable earlier. Let\u2019s see how differently it works using two variables.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167834556567\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831954242\">\r\n<div data-type=\"problem\" id=\"fs-id1167835174952\">\r\n<p id=\"fs-id1167830960764\">The sum of two numbers is zero. One number is nine less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835364658\">\r\n<table id=\"fs-id1167832010472\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for 2 numbers. Step 3 is to name the two numbers n and m. Step 4 is to translate into a system of equations. We get n plus m equal to 0 and n equal to m minus 9. Step 5 is to solve the system of equations. We substitute m minus n for n in the first equation and solve for m. We get m equal to 9 by 2. We substitute this in the second equation. Solving for n, we get n equal to minus 9 by 2. Step 6 is to check if the answer make sense in the problem. Step 7 is to answer the problem. The numbers are 9 by 2 and minus 9 by 2.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for two numbers.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(n=\\) the first number.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.4em}{0ex}}m=\\) the second number<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The sum of two numbers is zero.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835216620\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">One number is nine less than the other.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834395672\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is:<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832031159\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of\r\n<div data-type=\"newline\"><\/div>\r\nequations. We will use substitution\r\n<div data-type=\"newline\"><\/div>\r\nsince the second equation is solved\r\n<div data-type=\"newline\"><\/div>\r\nfor <em data-effect=\"italics\">n<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">m<\/em> \u2212 9 for <em data-effect=\"italics\">n<\/em> in the first equation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834094607\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">m<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826977660\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535269\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834525104\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute \\(m=\\frac{9}{2}\\) into the second equation\r\n<div data-type=\"newline\"><\/div>\r\nand then solve for <em data-effect=\"italics\">n<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835311214\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835201041\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835321418\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Do these numbers make sense in\r\n<div data-type=\"newline\"><\/div>\r\nthe problem? We will leave this to\r\n<div data-type=\"newline\"><\/div>\r\nyou!<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The numbers are \\(\\frac{9}{2}\\) and \\(-\\frac{9}{2}.\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835197823\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835634616\">\r\n<div data-type=\"problem\" id=\"fs-id1167835254462\">\r\n<p id=\"fs-id1167834228969\">The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835174756\">\r\n<p id=\"fs-id1167835174892\">3, 7<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826994044\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835390449\">\r\n<div data-type=\"problem\" id=\"fs-id1167835254405\">\r\n<p id=\"fs-id1167834515363\">The sum of two numbers is \\(-6.\\) One number is 10 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835378190\">\r\n\r\n2, \\(-8\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167835186838\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167830770008\">\r\n<div data-type=\"problem\" id=\"fs-id1167831872425\">\r\n\r\nHeather has been offered two options for her salary as a trainer at the gym. Option A would pay her ?25,000 plus ?15 for each training session. Option B would pay her \\(\\text{?}10,000+\\text{?}40\\) for each training session. How many training sessions would make the salary options equal?\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<table id=\"fs-id1167826994030\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the number of training sessions that would make the pay equal. Step 3 is to name what we are looking for. We name Heather\u2019s salary s and the number of training sessions n. Step 4 is to translate into a system of equations. Option A would pay her 25,000 dollars plus 15 dollars for each training session. So, s is 25,000 plus 15n. Option B would pay her 10,000 dollars plus 40 dollars for each training session. So, s is 10,000 plus 40n. Step 5 is to solve the system of equations. We substitute 25,000 plus 15n for s in the second equation and solve for n. We get n equal to 600. Step 6 is to check if the answer make sense in the problem. Are 600 training sessions a year reasonable? Are the two options equal when n equal to 600? Step 7 is to answer the problem. The salary options would be equal for 600 training sessions.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the number of\r\n<div data-type=\"newline\"><\/div>\r\ntraining sessions that would make\r\n<div data-type=\"newline\"><\/div>\r\nthe pay equal.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(s=\\) Heather\u2019s salary.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.5em}{0ex}}n=\\) the number of training sessions<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Option A would pay her ?25,000\r\n<div data-type=\"newline\"><\/div>\r\nplus ?15 for each training\r\n<div data-type=\"newline\"><\/div>\r\nsession.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835165797\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Option B would pay her ?10,000\r\n<div data-type=\"newline\"><\/div>\r\n+ ?40 for each training session.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792546807\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is shown.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826941234\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations.\r\n<div data-type=\"newline\"><\/div>\r\nWe will use substitution.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835332871\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute 25,000 +15<em data-effect=\"italics\">n<\/em> for <em data-effect=\"italics\">s<\/em> in the second\r\n<div data-type=\"newline\"><\/div>\r\nequation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305413\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826799100\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Are 600 training sessions a year reasonable?\r\n<div data-type=\"newline\"><\/div>\r\nAre the two options equal when <em data-effect=\"italics\">n<\/em> = 600?<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The salary options would be equal for 600 training\r\n<div data-type=\"newline\"><\/div>\r\nsessions.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835304408\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835366813\">\r\n<div data-type=\"problem\" id=\"fs-id1167835545400\">\r\n<p id=\"fs-id1167835335775\">Geraldine has been offered positions by two insurance companies. The first company pays a salary of ?12,000 plus a commission of ?100 for each policy sold. The second pays a salary of ?20,000 plus a commission of ?50 for each policy sold. How many policies would need to be sold to make the total pay the same?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830868501\">\r\n<p id=\"fs-id1167832068503\">160 policies<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835511221\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835303466\">\r\n<div data-type=\"problem\" id=\"fs-id1167835187907\">\r\n<p id=\"fs-id1167835237602\">Kenneth currently sells suits for company A at a salary of ?22,000 plus a ?10 commission for each suit sold. Company B offers him a position with a salary of ?28,000 plus a ?4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167835320702\">1000 suits<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168755200519\">As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167834064452\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167834464405\">\r\n<p id=\"fs-id1167831882418\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835231147\">When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes on the elliptical trainer and 30 minutes circuit training she burned 473 calories. How many calories does she burn for each minute on the elliptical trainer? How many calories for each minute of circuit training?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835320319\">\r\n<table id=\"fs-id1167835327644\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the number of calories burned each minute on the elliptical trainer and each minute of circuit training. Step 3 is to name what we are looking for. Let e be the number of calories burned per minute on the elliptical trainer and c be the number of calories burned per minute of circuit training. Step 4 is to translate into a system of equations. 10e plus 20c is 278 and 20e plus 30c is 473. Step 5 is to solve the system of equations. We multiply the first by minus 2 to get opposite coefficients of e. Simplify and add the equations and solve for c. We get c equal to 8.3. Substituting this value in either of the original equations, we get e equal to 11.2. Step 6 is to check the answer in the problem. Step 7 is to answer the problem. Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the number of\r\n<div data-type=\"newline\"><\/div>\r\ncalories burned each minute on the\r\n<div data-type=\"newline\"><\/div>\r\nelliptical trainer and each minute of\r\n<div data-type=\"newline\"><\/div>\r\ncircuit training.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(e=\\) number of calories burned per\r\n<div data-type=\"newline\"><\/div>\r\nminute on the elliptical trainer.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.4em}{0ex}}c=\\) number of calories burned per\r\n<div data-type=\"newline\"><\/div>\r\nminute while circuit training<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">10 minutes on the elliptical and circuit\r\n<div data-type=\"newline\"><\/div>\r\ntraining for 20 minutes, burned\r\n<div data-type=\"newline\"><\/div>\r\n278 calories<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305222\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">20 minutes on the elliptical and\r\n<div data-type=\"newline\"><\/div>\r\n30 minutes of circuit training burned\r\n<div data-type=\"newline\"><\/div>\r\n473 calories<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is:<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053714\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Multiply the first equation by \u22122 to get\r\n<div data-type=\"newline\"><\/div>\r\nopposite coefficients of <em data-effect=\"italics\">e<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832010294\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify and add the equations.\r\n<div data-type=\"newline\"><\/div>\r\nSolve for <em data-effect=\"italics\">c<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237274\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">c<\/em> = 8.3 into one of the\r\n<div data-type=\"newline\"><\/div>\r\noriginal equations to solve for <em data-effect=\"italics\">e<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831117433\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Check the math on your own.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509656\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Jenna burns 8.3 calories per minute\r\n<div data-type=\"newline\"><\/div>\r\ncircuit training and 11.2 calories per\r\n<div data-type=\"newline\"><\/div>\r\nminute while on the elliptical trainer.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834501534\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167830964293\">\r\n\r\nTranslate to a system of equations and then solve:\r\n\r\nMark went to the gym and did 40 minutes of Bikram hot yoga and 10 minutes of jumping jacks. He burned 510 calories. The next time he went to the gym, he did 30 minutes of Bikram hot yoga and 20 minutes of jumping jacks burning 470 calories. How many calories were burned for each minute of yoga? How many calories were burned for each minute of jumping jacks?\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167834064368\">Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831949038\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834346518\">\r\n<div data-type=\"problem\" id=\"fs-id1167835414903\">\r\n<p id=\"fs-id1167834539344\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835410359\">Erin spent 30 minutes on the rowing machine and 20 minutes lifting weights at the gym and burned 430 calories. During her next visit to the gym she spent 50 minutes on the rowing machine and 10 minutes lifting weights and burned 600 calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835334955\">\r\n\r\nErin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832152518\">\r\n<h3 data-type=\"title\">Solve Geometry Applications<\/h3>\r\n<p id=\"fs-id1167835241391\">We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.<\/p>\r\n<p id=\"fs-id1167830702417\">The measures of two <span data-type=\"term\">complementary angles<\/span> add to 90 degrees. The measures of two <span data-type=\"term\">supplementary angles<\/span> add to 180 degrees.<\/p>\r\n\r\n<div data-type=\"note\">\r\n<div data-type=\"title\">Complementary and Supplementary Angles<\/div>\r\n<p id=\"fs-id1167835514605\">Two angles are complementary if the sum of the measures of their angles is 90 degrees.<\/p>\r\n<p id=\"fs-id1167835329300\">Two angles are supplementary if the sum of the measures of their angles is 180 degrees.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167835200476\">If two angles are complementary, we say that <em data-effect=\"italics\">one angle is the complement of the other.<\/em><\/p>\r\n<p id=\"fs-id1167835201883\">If two angles are supplementary, we say that <em data-effect=\"italics\">one angle is the supplement of the other.<\/em><\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835363488\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167831228660\">\r\n<p id=\"fs-id1167832153433\">Translate to a system of equations and then solve.<\/p>\r\n<p id=\"fs-id1167834059688\">The difference of two complementary angles is 26 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835281517\">\r\n<p id=\"fs-id1167835371347\">\\(\\begin{array}{cccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what we are looking for.}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{We are looking for the measure of each}\\hfill \\\\ \\text{angle.}\\hfill \\end{array}\\hfill \\\\ \\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what we are looking for.}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=\\phantom{\\rule{0.2em}{0ex}}\\text{the measure of the first angle.}\\hfill \\\\ \u2003\u2003y=\\phantom{\\rule{0.2em}{0ex}}\\text{the measure of the second angle}\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 4. Translate}}\\phantom{\\rule{0.2em}{0ex}}\\text{into a system of}\\hfill \\\\ \\text{equations.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{The angles are complementary.}\\hfill \\\\ \\phantom{\\rule{5.5em}{0ex}}x+y=90\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; \\begin{array}{c}\\text{The difference of the two angles is 26}\\hfill \\\\ \\text{degrees.}\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{5.6em}{0ex}}x-y=26\\hfill \\\\ \\\\ \\\\ \\text{The system is shown.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5.3em}{0ex}}\\left\\{\\begin{array}{c}x+y=90\\hfill \\\\ x-y=26\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the system of equations}\\hfill \\\\ \\text{by elimination.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\begin{array}{c}\\underset{___________}{\\left\\{\\begin{array}{c}x+y=90\\hfill \\\\ x-y=26\\hfill \\end{array}}\\hfill \\\\ \u00a02x\\phantom{\\rule{1em}{0ex}}=116\\hfill \\end{array}\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}x=58\\phantom{\\rule{0.2em}{0ex}}\\text{into the first equation.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5.2em}{0ex}}\\begin{array}{c}\\hfill x=58\\\\ \\\\ \\\\ \\hfill x+y=90\\\\ \\hfill 58+y=90\\\\ \\hfill y=32\\end{array}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem.}\\hfill \\\\ \\\\ \\\\ \\\\ \\phantom{\\rule{4em}{0ex}}\\begin{array}{c}58+32=90\u2713\\hfill \\\\ 58-32=26\u2713\\hfill \\end{array}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 7. Answer}}\\phantom{\\rule{0.2em}{0ex}}\\text{the question.}\\hfill &amp; &amp; &amp; \\text{The angle measures are 58 and 32 degrees.}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831985764\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167834195982\">\r\n<p id=\"fs-id1167835544861\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167834192154\">The difference of two complementary angles is 20 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167831922064\">The angle measures are 55 and 35.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835365592\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835346376\">\r\n<div data-type=\"problem\" id=\"fs-id1167835355260\">\r\n<p id=\"fs-id1167835419750\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835330289\">The difference of two complementary angles is 80 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835390037\">\r\n\r\nThe angle measures are 5 and 85.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168756438299\">In the next example, we remember that the measures of supplementary angles add to 180.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835342807\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831239714\">\r\n<div data-type=\"problem\" id=\"fs-id1167835360777\">\r\n<p id=\"fs-id1167830963487\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167826781076\">Two angles are supplementary. The measure of the larger angle is twelve degrees less than five times the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834194375\">\r\n<table id=\"fs-id1167835325851\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the measure of each angle. Step 3 is to name what we are looking for. Let x be the measure of the first angle and y be the measure of the second angle. Step 4 is to translate into a system of equations. The angles are supplementary. So, x plus y is 180 and y is 5 x minus 12. Step 5 is to solve the system of equations by substitution. Substitute 5 x minus 12 for y in the first equation. Solving for x, we get x equal to 32. Substitute this in the second equation and solve for y. We get y equal to 148. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The angle measures are 148 and 32.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for measure of each\r\n<div data-type=\"newline\"><\/div>\r\nangle.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(x=\\) the measure of the first angle.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.5em}{0ex}}y=\\) the measure of the second angle<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The angles are supplementary.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064608\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The larger angle is twelve less than five\r\n<div data-type=\"newline\"><\/div>\r\ntimes the smaller angle.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834097914\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is shown:\r\n<div data-type=\"newline\"><\/div>\r\n<strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations substitution.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831112459\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute 5<em data-effect=\"italics\">x<\/em> \u2212 12 for <em data-effect=\"italics\">y<\/em> in the first equation.\r\n<div data-type=\"newline\"><\/div>\r\nSolve for <em data-effect=\"italics\">x<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390372\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nSubstitute 32 for <em data-effect=\"italics\">x<\/em> in the second\r\n<div data-type=\"newline\"><\/div>\r\nequation, then solve for <em data-effect=\"italics\">y<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336028\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167830991319\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834394937\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The angle measures are 148 and 32 degrees.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835217900\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831887926\">\r\n<div data-type=\"problem\" id=\"fs-id1167830960497\">\r\n<p id=\"fs-id1167835335238\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167826783686\">Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835420011\">\r\n<p id=\"fs-id1167835344064\">The angle measures are 42 and 138.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826778516\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167835390453\">\r\n<p id=\"fs-id1167834429847\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835329416\">Two angles are supplementary. The measure of the larger angle is 18 less than twice the measure of the smaller angle. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834184127\">\r\n<p id=\"fs-id1167831970277\">The angle measures are 66 and 114.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835280307\">Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835167148\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835367895\">\r\n<div data-type=\"problem\" id=\"fs-id1167835384327\">\r\n\r\n4.19\r\n<p id=\"fs-id1167831881234\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831822903\">\r\n<p id=\"fs-id1167834583384\">We will draw and label a figure.<\/p>\r\n\r\n<table id=\"fs-id1167831884545\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the measures of the two angles of the triangle. Step 3 is to name what we are looking for. Let A equal the measure of the first angle. Let B equal the measure of the second angle. Step 4 is to translate into a system of equations. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. This means that A equals 3 B plus 10. The sum of the measures of the angles of a triangle is 180. A plus B plus 90 equals 180. The system is A equals 3 B plus 10 and A plus B plus 90 equals 180. Step 5 is to solve the system of equations. We will use substitution since the first equation is solved for A. Because A equals 3 B plus 10, we substitute 3 B plus 10 in for A in the second equation. This gives us 3 B plus 10, in parentheses, plus B plus 90 equals 180. Solve for B. 4 B plus 100 equals 180. 4 B equals 80. B equals 20. Substitute B equals 20 into the first equation and then solve for A. A equals 3 times 20 plus 10. A equals 70. Step 6 is to check the answer in the problem. We will leave this to you. Step 7 is to answer the question. The measures of the small angles are 20 and 70.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832065615\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the measures of the angles.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(a=\\) the measure of the first angle.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.5em}{0ex}}b=\\) the measure of the second angle<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The sum of the measures of the angles of a triangle is 180.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835279918\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is shown.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835337274\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations. We will use substitution since the first equation is solved for <em data-effect=\"italics\">a<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835640016\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute \\(3b+10\\) for <em data-effect=\"italics\">a<\/em> in the second equation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835202914\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">b<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254030\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute \\(b=20\\) into the first equation and then solve for <em data-effect=\"italics\">a.<\/em><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834063905\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We will leave this to you!<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The measures of the small angles are 20 and 70 degrees.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834066299\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834402920\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167831958101\">The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167830705915\">\r\n<p id=\"fs-id1167834403373\">22, 68<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834469592\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834194118\">\r\n<div data-type=\"problem\" id=\"fs-id1167826874759\">\r\n<p id=\"fs-id1167834228931\">The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826802370\">\r\n<p id=\"fs-id1167834532312\">36, 54<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168756447619\">Often it is helpful when solving geometry applications to draw a picture to visualize the situation.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835231793\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835254341\">\r\n<div data-type=\"problem\" id=\"fs-id1167835306773\">\r\n<p id=\"fs-id1167835380024\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835483610\">Randall has 125 feet of fencing to enclose the part of his backyard adjacent to his house. He will only need to fence around three sides, because the fourth side will be the wall of the house. He wants the length of the fenced yard (parallel to the house wall) to be 5 feet more than four times as long as the width. Find the length and the width.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834064239\">\r\n<table id=\"fs-id1167832054759\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the length and width. Step 3 is to name what we are looking for. Let L be the length and W be the width. Step 4 is to translate into a system of equations. L plus 2W is 125 and L is 4W plus 5. Step 5 is to solve the system of equations. Substituting the second equation into the first and solving for W, we get W equal to 20. Substituting this into the second equation, we get L equal to 85. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The length is 85 feet and width is 20 feet.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the length and width.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237826\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(L=\\) the length of the fenced yard.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.3em}{0ex}}W=\\) the width of the fenced yard<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">One lenth and two widths equal 125.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834395872\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The length will be 5 feet more than\r\n<div data-type=\"newline\"><\/div>\r\nfour times the width.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The system is shown.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<strong data-effect=\"bold\">Step 5. Solve<\/strong> The system of equations\r\n<div data-type=\"newline\"><\/div>\r\nby substitution.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826994454\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">L<\/em> = 4<em data-effect=\"italics\">W<\/em> + 5 into the first\r\n<div data-type=\"newline\"><\/div>\r\nequation, then solve for <em data-effect=\"italics\">W<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358000\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345478\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute 20 for <em data-effect=\"italics\">W<\/em> in the second\r\n<div data-type=\"newline\"><\/div>\r\nequation, then solve for <em data-effect=\"italics\">L<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the\r\n<div data-type=\"newline\"><\/div>\r\nproblem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834598458\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the equation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The length is 85 feet and the width is 20 feet.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167830697622\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834191195\">\r\n<div data-type=\"problem\" id=\"fs-id1167834300837\">\r\n<p id=\"fs-id1167831921373\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167832066931\">Mario wants to put a fence around the pool in his backyard. Since one side is adjacent to the house, he will only need to fence three sides. There are two long sides and the one shorter side is parallel to the house. He needs 155 feet of fencing to enclose the pool. The length of the long side is 10 feet less than twice the width. Find the length and width of the pool area to be enclosed.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835367103\">\r\n<p id=\"fs-id1167835498160\">The length is 60 feet and the width is 35 feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832066759\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831881110\">\r\n<div data-type=\"problem\" id=\"fs-id1167835300576\">\r\n<p id=\"fs-id1167828433171\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167834151997\">Alexis wants to build a rectangular dog run in her yard adjacent to her neighbor\u2019s fence. She will use 136 feet of fencing to completely enclose the rectangular dog run. The length of the dog run along the neighbor\u2019s fence will be 16 feet less than twice the width. Find the length and width of the dog run.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835368759\">\r\n<p id=\"fs-id1167832058855\">The length is 60 feet and the width is 38 feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835180813\">\r\n<h3 data-type=\"title\">Solve uniform motion applications<\/h3>\r\n<p id=\"fs-id1167828421536\">We used a table to organize the information in uniform motion problems when we introduced them earlier. We\u2019ll continue using the table here. The basic equation was \\(D=rt\\) where <em data-effect=\"italics\">D<\/em> is the distance traveled, <em data-effect=\"italics\">r<\/em> is the rate, and <em data-effect=\"italics\">t<\/em> is the time.<\/p>\r\n<p id=\"fs-id1167834279160\">Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.<\/p>\r\n\r\n<div data-type=\"example\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167831920196\">\r\n<div data-type=\"problem\" id=\"fs-id1167835304306\">\r\n<p id=\"fs-id1167835351018\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167828240732\">Joni left St. Louis on the interstate, driving west towards Denver at a speed of 65 miles per hour. Half an hour later, Kelly left St. Louis on the same route as Joni, driving 78 miles per hour. How long will it take Kelly to catch up to Joni?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835261754\">\r\n<p id=\"fs-id1167834229330\">A diagram is useful in helping us visualize the situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167834147077\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n<p id=\"fs-id1167830700663\"><strong data-effect=\"bold\">Identify and name<\/strong> what we are looking for. A chart will help us organize the data. We know the rates of both Joni and Kelly, and so we enter them in the chart. We are looking for the length of time Kelly, <em data-effect=\"italics\">k<\/em>, and Joni, <em data-effect=\"italics\">j<\/em>, will each drive.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167826996003\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n<p id=\"fs-id1167834189936\">Since \\(D=r\u00b7t\\) we can fill in the Distance column.<\/p>\r\n<p id=\"fs-id1167835513925\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/p>\r\n<p id=\"fs-id1167835308664\">To make the system of equations, we must recognize that Kelly and Joni will drive the same distance. So,<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1169147949384\" class=\"unnumbered\" data-label=\"\">\\(\\phantom{\\rule{5em}{0ex}}65j=78k\\)<\/div>\r\n<p id=\"fs-id1167834473791\">Also, since Kelly left later, her time will be \\(\\frac{1}{2}\\) hour less than Joni\u2019s time. So,<\/p>\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}k=j-\\frac{1}{2}\\hfill \\\\ \\\\ \\\\ \\text{Now we have the system.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.4em}{0ex}}\\left\\{\\begin{array}{c}\\phantom{\\rule{1em}{0ex}}k=j-\\frac{1}{2}\\hfill \\\\ 65j=78k\\hfill \\end{array}\\hfill \\\\ \\mathbf{\\text{Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the system of equations by substitution.}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{}\\\\ \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}k=j-\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\\text{into the second equation,}\\hfill \\\\ \\text{then solve for}\\phantom{\\rule{0.2em}{0ex}}j.\\hfill \\end{array}\\hfill &amp; &amp; \\\\ &amp; &amp; &amp; \\phantom{\\rule{3.2em}{0ex}}65j=78k\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{2.1em}{0ex}}\\begin{array}{ccc}\\hfill 65j&amp; =\\hfill &amp; 78\\left(j-\\frac{1}{2}\\right)\\hfill \\\\ \\hfill 65j&amp; =\\hfill &amp; 78j-39\\hfill \\\\ \\hfill -13j&amp; =\\hfill &amp; -39\\hfill \\\\ \\hfill j&amp; =\\hfill &amp; 3\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\text{To find Kelly\u2019s time, substitute}\\phantom{\\rule{0.2em}{0ex}}j=3\\phantom{\\rule{0.2em}{0ex}}\\text{into the first}\\hfill \\\\ \\text{equation, then solve for}\\phantom{\\rule{0.2em}{0ex}}k.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4.3em}{0ex}}k=j-\\frac{1}{2}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4.3em}{0ex}}k=3-\\frac{1}{2}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{1em}{0ex}}k=\\frac{5}{2}\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}k=2\\frac{1}{2}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer in the problem.}\\hfill \\\\ \\text{Joni}\\phantom{\\rule{1.8em}{0ex}}3\\phantom{\\rule{0.2em}{0ex}}\\text{hours}\\phantom{\\rule{0.2em}{0ex}}\\left(65\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\right)\\phantom{\\rule{.2em}{0ex}}=195\\phantom{\\rule{0.2em}{0ex}}\\text{miles}\\hfill \\\\ \\text{Kelly}\\phantom{\\rule{0.5em}{0ex}}2\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\\text{hours}\\phantom{\\rule{0.2em}{0ex}}\\left(78\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\right)=195\\phantom{\\rule{0.2em}{0ex}}\\text{miles}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\text{Yes, they will have traveled the same distance}\\hfill \\\\ \\text{when they meet.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\mathbf{\\text{Answer}}\\phantom{\\rule{0.2em}{0ex}}\\text{the question.}\\hfill \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Kelly will catch up to Joni in}\\hfill \\\\ 2\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\\text{hours. By then, Joni will}\\hfill \\\\ \\text{have traveled 3 hours.}\\hfill \\end{array}\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831970009\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167832052250\">\r\n<p id=\"fs-id1167834186157\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167834059465\">Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834053473\">\r\n<p id=\"fs-id1167834111767\">It will take Clark 4 hours to catch Mitchell.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167831883648\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834396468\">\r\n<div data-type=\"problem\" id=\"fs-id1167835533572\">\r\n<p id=\"fs-id1167834432254\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167831880362\">Charlie left his mother\u2019s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes \\(\\left(\\frac{1}{4}\\phantom{\\rule{0.2em}{0ex}}\\text{hour}\\right)\\) later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835320770\">\r\n<p id=\"fs-id1167826783003\">It will take Sally \\(1\\frac{1}{2}\\) hours to catch up to Charlie.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167835306358\">Many real-world applications of uniform motion arise because of the effects of currents\u2014of water or air\u2014on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.<\/p>\r\n<p id=\"fs-id1167835191090\">Let\u2019s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.<\/p>\r\n<p id=\"fs-id1167834472600\">The images below show how a river current affects the speed at which a boat is actually travelling. We\u2019ll call the speed of the boat in still water <em data-effect=\"italics\">b<\/em> and the speed of the river current <em data-effect=\"italics\">c<\/em>.<\/p>\r\n<p id=\"fs-id1167834098208\">The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat\u2019s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is \\(b+c.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167827958519\" data-alt=\"Figure shows a boat and two horizontal arrows, both pointing left. The one to the left of the boat is b and the one to the right is c.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a boat and two horizontal arrows, both pointing left. The one to the left of the boat is b and the one to the right is c.\" \/><\/span>\r\n<p id=\"fs-id1167835181916\">Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat\u2019s actual speed is slower than its speed in still water. The actual speed of the boat is \\(b-c.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167835610140\" data-alt=\"Figure shows a boat and two horizontal arrows to its left. One, labeled b, points left and the other, labeled c, points right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a boat and two horizontal arrows to its left. One, labeled b, points left and the other, labeled c, points right.\" \/><\/span>\r\n<p id=\"fs-id1167835333178\">We\u2019ll put some numbers to this situation in the next example.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167835534188\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835238926\">\r\n<div data-type=\"problem\" id=\"fs-id1167834053490\">\r\n<p id=\"fs-id1167835365522\">Translate to a system of equations and then solve.<\/p>\r\n<p id=\"fs-id1167835356872\">A river cruise ship sailed 60 miles downstream for 4 hours and then took 5 hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831883744\">\r\n<table id=\"fs-id1167835343141\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. This is a uniform motion problem and a picture will help us visualize the situation. The picture shows a wavy line pointing right. This is labeled c. A straight arrow pointing right is labeled 4 hours. Another arrow pointing left is labeled 5 hours. The length of arrows is labeled 60 miles. Step 2 is to identify what we are looking for. We are looking for the speed of the ship in still water and the speed of the current. Step 3 is to name what we are looking for. Let s be the rate of ship in still water and c be the rate of current. The ship goes downstream and then upstream. Going downstream, the current helps the ship and so the ship's actual rate is s plus c. Going upstream, the current slows the ship and so the actual rate is s minus c. Step 4 is to translate into a system of equations. 4 open parentheses s plus c close parentheses is 60 and 5 open parentheses s minus c close parentheses is 60. Step 5 is to solve the system of equations. Multiply the top equation by 5 and the bottom one by 4. By adding them and solving for s, we get s equal to 13.5. By substituting, we get c equal to 1.5. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The rate of ship is 13.5 and rate of current is 1.5.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">This is a uniform motion problem and a\r\n<div data-type=\"newline\"><\/div>\r\npicture will help us visualize the situation.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835341268\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the speed of the ship\r\n<div data-type=\"newline\"><\/div>\r\nin still water and the speed of the current.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(s=\\) the rate of the ship in still water.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.5em}{0ex}}c=\\) the rate of the current<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">A chart will help us organize the information.\r\n<div data-type=\"newline\"><\/div>\r\nThe ship goes downstream and then upstream.\r\n<div data-type=\"newline\"><\/div>\r\nGoing downstream, the current helps the\r\n<div data-type=\"newline\"><\/div>\r\nship and so the ship's actual rate is <em data-effect=\"italics\">s<\/em> + <em data-effect=\"italics\">c<\/em>.\r\n<div data-type=\"newline\"><\/div>\r\nGoing upstream, the current slows the ship\r\n<div data-type=\"newline\"><\/div>\r\nand so the actual rate is <em data-effect=\"italics\">s<\/em> \u2212 <em data-effect=\"italics\">c<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556600\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Downstream it takes 4 hours.\r\n<div data-type=\"newline\"><\/div>\r\nUpstream it takes 5 hours.\r\n<div data-type=\"newline\"><\/div>\r\nEach way the distance is 60 miles.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.\r\n<div data-type=\"newline\"><\/div>\r\nSince rate times time is distance, we can\r\n<div data-type=\"newline\"><\/div>\r\nwrite the system of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the system of equations.\r\n<div data-type=\"newline\"><\/div>\r\nDistribute to put both equations in standard\r\n<div data-type=\"newline\"><\/div>\r\nform, then solve by elimination.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327540\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Multiply the top equation by 5 and the\r\n<div data-type=\"newline\"><\/div>\r\nbottom equation by 4.\r\n<div data-type=\"newline\"><\/div>\r\nAdd the equations, then solve for <em data-effect=\"italics\">s<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835361923\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">s<\/em> = 13.5 into of the original\r\n<div data-type=\"newline\"><\/div>\r\nequations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835512731\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Check<\/strong> the answer in the problem.\r\n<div data-type=\"newline\"><\/div>\r\nThe downstream rate would be\r\n<div data-type=\"newline\"><\/div>\r\n\\(13.5+1.5=15\\) mph.\r\n<div data-type=\"newline\"><\/div>\r\nIn 4 hours the ship would travel\r\n<div data-type=\"newline\"><\/div>\r\n\\(15\u00b74=60\\) miles.\r\n<div data-type=\"newline\"><\/div>\r\nThe upstream rate would be\r\n<div data-type=\"newline\"><\/div>\r\n\\(13.5-1.5=12\\) mph.\r\n<div data-type=\"newline\"><\/div>\r\nIn 5 hours the ship would travel\r\n<div data-type=\"newline\"><\/div>\r\n\\(12\u00b75=60\\) miles.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The rate of the ship is 13.5 mph and\r\n<div data-type=\"newline\"><\/div>\r\nthe rate of the current is 1.5 mph.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834099440\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834183359\">\r\n<div data-type=\"problem\" id=\"fs-id1167831832521\">\r\n<p id=\"fs-id1167834183093\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167831955755\">A Mississippi river boat cruise sailed 120 miles upstream for 12 hours and then took 10 hours to return to the dock. Find the speed of the river boat in still water and the speed of the river current.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834227436\">\r\n<p id=\"fs-id1167834367202\">The rate of the boat is 11 mph and the rate of the current is 1 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167834314564\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167834279563\">\r\n<div data-type=\"problem\" id=\"fs-id1167835349130\">\r\n<p id=\"fs-id1167830909024\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835284768\">Jason paddled his canoe 24 miles upstream for 4 hours. It took him 3 hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826997660\">\r\n<p id=\"fs-id1167835352364\">The speed of the canoe is 7 mph and the speed of the current is 1 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167834432815\">Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We\u2019ll see this in the next example. A wind current in the same direction as the plane is flying is called a <em data-effect=\"italics\">tailwind<\/em>. A wind current blowing against the direction of the plane is called a <em data-effect=\"italics\">headwind<\/em>.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167831836495\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832015630\">\r\n<div data-type=\"problem\" id=\"fs-id1167834284501\">\r\n<p id=\"fs-id1167835338954\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167835327022\">A private jet can fly 1,095 miles in three hours with a tailwind but only 987 miles in three hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835319175\">\r\n<table id=\"fs-id1167834327609\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. This is a uniform motion problem and a picture will help us visualize the situation. The picture shows a wavy line pointing right. This is labeled wind, 3 hours. A straight arrow pointing right is labeled j plus w equals 365, 1095 miles. Another arrow, pointing left is labeled j minus w equals 329, 987 miles. Step 2 is to identify what we are looking for. We are looking for the speed of the jet in still air and the speed of the wind. Step 3 is to name what we are looking for. Let j be the speed of jet in still air and w be the speed of wind. Step 4 is to translate into a system of equations. 3j plus 3w is 1095 and 3j minus 3w is 987. Step 5 is to solve the system of equations. Solving by elimination, we get j equal to 347. Solving by substitution we get w equal to 18. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The rate of the jet is 347 mph and rate of wind is 18 mph.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">This is a uniform motion problem and a\r\n<div data-type=\"newline\"><\/div>\r\npicture will help us visualize.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327739\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">We are looking for the speed of the jet\r\n<div data-type=\"newline\"><\/div>\r\nin still air and the speed of the wind.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> what we are looking for.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">Let \\(j=\\) the speed of the jet in still air.\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.3em}{0ex}}w=\\) the speed of the wind.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">A chart will help us organize the information.\r\n<div data-type=\"newline\"><\/div>\r\nThe jet makes two trips\u2014one in a tailwind\r\n<div data-type=\"newline\"><\/div>\r\nand one in a headwind.\r\n<div data-type=\"newline\"><\/div>\r\nIn a tailwind, the wind helps the jet and so\r\n<div data-type=\"newline\"><\/div>\r\nthe rate is <em data-effect=\"italics\">j<\/em> + <em data-effect=\"italics\">w<\/em>.\r\n<div data-type=\"newline\"><\/div>\r\nIn a headwind, the wind slows the jet and\r\n<div data-type=\"newline\"><\/div>\r\nso the rate is <em data-effect=\"italics\">j<\/em> \u2012 <em data-effect=\"italics\">w<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834397046\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Each trip takes 3 hours.\r\n<div data-type=\"newline\"><\/div>\r\nIn a tailwind the jet flies 1,095 miles.\r\n<div data-type=\"newline\"><\/div>\r\nIn a headwind the jet flies 987 miles.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.\r\n<div data-type=\"newline\"><\/div>\r\nSince rate times time is distance, we get the\r\n<div data-type=\"newline\"><\/div>\r\nsystem of equations.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826941282\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the system of equations.\r\n<div data-type=\"newline\"><\/div>\r\nDistribute, then solve by elimination.\r\n<div data-type=\"newline\"><\/div>\r\nAdd, and solve for <em data-effect=\"italics\">j<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831890692\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">j<\/em> = 347 into one of the original\r\n<div data-type=\"newline\"><\/div>\r\nequations, then solve for <em data-effect=\"italics\">w<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835341957\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Check<\/strong> the answer in the problem.\r\n<div data-type=\"newline\"><\/div>\r\nWith the tailwind, the actual rate of the\r\n<div data-type=\"newline\"><\/div>\r\njet would be\r\n<div data-type=\"newline\"><\/div>\r\n\\(347+18=365\\) mph.\r\n<div data-type=\"newline\"><\/div>\r\nIn 3 hours the jet would travel\r\n<div data-type=\"newline\"><\/div>\r\n\\(365\u00b73=1,095\\) miles\r\n<div data-type=\"newline\"><\/div>\r\nGoing into the headwind, the jet\u2019s actual\r\n<div data-type=\"newline\"><\/div>\r\nrate would be\r\n<div data-type=\"newline\"><\/div>\r\n\\(347-18=329\\) mph.\r\n<div data-type=\"newline\"><\/div>\r\nIn 3 hours the jet would travel\r\n<div data-type=\"newline\"><\/div>\r\n\\(329\u00b73=987\\) miles.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The rate of the jet is 347 mph and the\r\n<div data-type=\"newline\"><\/div>\r\nrate of the wind is 18 mph.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167835233774\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167835360362\">\r\n<div data-type=\"problem\" id=\"fs-id1167835368111\">\r\n<p id=\"fs-id1167834556843\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167834226271\">A small jet can fly 1,325 miles in 5 hours with a tailwind but only 1,035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835345606\">\r\n<p id=\"fs-id1167835234036\">The speed of the jet is 235 mph and the speed of the wind is 30 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826998350\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167828435014\">\r\n<div data-type=\"problem\" id=\"fs-id1167834345458\">\r\n<p id=\"fs-id1167826977977\">Translate to a system of equations and then solve:<\/p>\r\n<p id=\"fs-id1167834462714\">A commercial jet can fly 1,728 miles in 4 hours with a tailwind but only 1,536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835322254\">\r\n<p id=\"fs-id1167835231016\">The speed of the jet is 408 mph and the speed of the wind is 24 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"media-2\">\r\n<p id=\"fs-id1167832151228\">Access this online resource for additional instruction and practice with systems of equations.<\/p>\r\n\r\n<ul id=\"fs-id1169148234280\" data-display=\"block\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37sysequations\">Systems of Equations<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831921394\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167831881094\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">How To Solve Applications with Systems of Equations<\/strong>\r\n<ol id=\"fs-id1167835369775\" class=\"stepwise\" type=\"1\">\r\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\r\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\r\n \t<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\r\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\r\n \t<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\r\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835364750\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834141990\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1167835207310\"><strong data-effect=\"bold\">Direct Translation Applications<\/strong><\/p>\r\n<p id=\"fs-id1167835344124\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167834535347\">\r\n<div data-type=\"problem\" id=\"fs-id1167826813970\">\r\n<p id=\"fs-id1167835348193\">The sum of two number is 15. One number is 3 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167832055884\">\r\n<p id=\"fs-id1167835534165\">The sum of two number is 30. One number is 4 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835434156\">\r\n\r\n13 and 17\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832057971\">\r\n<div data-type=\"problem\" id=\"fs-id1167832056216\">\r\n<p id=\"fs-id1167831910371\">The sum of two number is \u221216. One number is 20 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834395267\">\r\n<div data-type=\"problem\" id=\"fs-id1167832068556\">\r\n<p id=\"fs-id1167831880227\">The sum of two number is \\(-26.\\) One number is 12 less than the other. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835361560\">\r\n<p id=\"fs-id1167835253917\">\\(-7\\) and \\(-19\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834527974\">\r\n<div data-type=\"problem\" id=\"fs-id1167835339583\">\r\n<p id=\"fs-id1167831919812\">The sum of two numbers is 65. Their difference is 25. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835382125\">\r\n<div data-type=\"problem\" id=\"fs-id1167830696690\">\r\n<p id=\"fs-id1167835518468\">The sum of two numbers is 37. Their difference is 9. Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835342748\">\r\n<p id=\"fs-id1167835378005\">14 and 23<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834319791\">\r\n<div data-type=\"problem\" id=\"fs-id1167834053763\">\r\n<p id=\"fs-id1167835310165\">The sum of two numbers is \\(-27.\\) Their difference is \\(-59.\\) Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835350540\">\r\n<div data-type=\"problem\" id=\"fs-id1167834138030\">\r\n<p id=\"fs-id1167835349416\">The sum of two numbers is \\(-45.\\) Their difference is \\(-89.\\) Find the numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835421990\">\r\n<p id=\"fs-id1167835420760\">22 and \\(-67\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831824722\">\r\n<div data-type=\"problem\" id=\"fs-id1167835483590\">\r\n<p id=\"fs-id1167835381769\">Maxim has been offered positions by two car companies. The first company pays a salary of ?10,000 plus a commission of ?1000 for each car sold. The second pays a salary of ?20,000 plus a commission of ?500 for each car sold. How many cars would need to be sold to make the total pay the same?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834473411\">\r\n<div data-type=\"problem\" id=\"fs-id1167834516218\">\r\n<p id=\"fs-id1167831162894\">Jackie has been offered positions by two cable companies. The first company pays a salary of ?14,000 plus a commission of ?100 for each cable package sold. The second pays a salary of ?20,000 plus a commission of ?25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832066666\">\r\n\r\nEighty cable packages would need to be sold to make the total pay the same.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834238729\">\r\n<div data-type=\"problem\" id=\"fs-id1167835628567\">\r\n<p id=\"fs-id1167835609615\">Amara currently sells televisions for company A at a salary of ?17,000 plus a ?100 commission for each television she sells. Company B offers her a position with a salary of ?29,000 plus a ?20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834085038\">\r\n<div data-type=\"problem\" id=\"fs-id1167834085040\">\r\n<p id=\"fs-id1167835509988\">Mitchell currently sells stoves for company A at a salary of ?12,000 plus a ?150 commission for each stove he sells. Company B offers him a position with a salary of ?24,000 plus a ?50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835527846\">\r\n<p id=\"fs-id1167835253891\">Mitchell would need to sell 120 stoves for the companies to be equal.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831920428\">\r\n<div data-type=\"problem\" id=\"fs-id1167831920430\">\r\n<p id=\"fs-id1167832153327\">Two containers of gasoline hold a total of fifty gallons. The big container can hold ten gallons less than twice the small container. How many gallons does each container hold?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826788597\">\r\n<div data-type=\"problem\" id=\"fs-id1167834246873\">\r\n<p id=\"fs-id1167826828264\">June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826880014\">\r\n<p id=\"fs-id1167826880016\">8 and 40 gallons<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834395746\">\r\n<div data-type=\"problem\" id=\"fs-id1167835479730\">\r\n<p id=\"fs-id1167835479732\">Shelly spent 10 minutes jogging and 20 minutes cycling and burned 300 calories. The next day, Shelly swapped times, doing 20 minutes of jogging and 10 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834130681\">\r\n<div data-type=\"problem\" id=\"fs-id1167835361143\">\r\n<p id=\"fs-id1167835361146\">Drew burned 1800 calories Friday playing one hour of basketball and canoeing for two hours. Saturday he spent two hours playing basketball and three hours canoeing and burned 3200 calories. How many calories did he burn per hour when playing basketball? How many calories did he burn per hour when canoeing?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826987517\">\r\n<p id=\"fs-id1167826987519\">1000 calories playing basketball and 400 calories canoeing<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835353325\">\r\n<div data-type=\"problem\" id=\"fs-id1167831036987\">\r\n<p id=\"fs-id1167831036989\">Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for ?116. Lisa bought two notebooks and three thumb dives for ?68. Find the cost of each notebook and each thumb drive.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832053097\">\r\n<div data-type=\"problem\" id=\"fs-id1167832066087\">\r\n<p id=\"fs-id1167832066089\">Nancy bought seven pounds of oranges and three pounds of bananas for ?17. Her husband later bought three pounds of oranges and six pounds of bananas for ?12. What was the cost per pound of the oranges and the bananas?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832053548\">\r\n<p id=\"fs-id1167830898572\">Oranges cost ?2 per pound and bananas cost ?1 per pound<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834535330\">\r\n<div data-type=\"problem\" id=\"fs-id1167834535332\">\r\n<p id=\"fs-id1167834367247\">Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for ?114 or she is able to buy 2 shirts and 4 sweaters for ?164. How much does a shirt cost? How much does a sweater cost?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834422585\">\r\n<div data-type=\"problem\" id=\"fs-id1167832153537\">\r\n<p id=\"fs-id1167834539431\">Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for ?40 or he is able to buy 5 packages of paper and 6 staplers for ?62. How much does a package of paper cost? How much does a stapler cost?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167827943703\">\r\n<p id=\"fs-id1167827943705\">Package of paper ?4, stapler ?7<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835374673\">\r\n<div data-type=\"problem\" id=\"fs-id1167831922575\">\r\n<p id=\"fs-id1167831922577\">The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830699166\">\r\n<div data-type=\"problem\" id=\"fs-id1167830699168\">\r\n<p id=\"fs-id1167826940670\">The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835234228\">\r\n<p id=\"fs-id1167835354843\">Hot dog 150 calories, cup of cottage cheese 220 calories<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834403211\">\r\n<div data-type=\"problem\" id=\"fs-id1167835309435\">\r\n<p id=\"fs-id1167835417541\">Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water as juice. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831880334\">\r\n<div data-type=\"problem\" id=\"fs-id1167834191245\">\r\n<p id=\"fs-id1167834438919\">Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835360066\">\r\n<p id=\"fs-id1167835360068\">Owen will need 80 quarts of water and 20 quarts of concentrate to make 100 quarts of lemonade.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167831871616\"><strong data-effect=\"bold\">Solve Geometry Applications<\/strong><\/p>\r\n<p id=\"fs-id1167835533545\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167832042398\">\r\n<div data-type=\"problem\" id=\"fs-id1167832042400\">\r\n\r\nThe difference of two complementary angles is 55 degrees. Find the measures of the angles.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835530147\">\r\n<div data-type=\"problem\" id=\"fs-id1167835530149\">\r\n<p id=\"fs-id1167834192549\">The difference of two complementary angles is 17 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834079388\">\r\n<p id=\"fs-id1167831811616\">\\(53.5\\) degrees and \\(36.5\\) degrees<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831908441\">\r\n<div data-type=\"problem\" id=\"fs-id1167831908443\">\r\n<p id=\"fs-id1167832066177\">Two angles are complementary. The measure of the larger angle is twelve less than twice the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835512309\">\r\n<div data-type=\"problem\" id=\"fs-id1167834396369\">\r\n<p id=\"fs-id1167834396372\">Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835281915\">\r\n<p id=\"fs-id1167835229140\">16 degrees and 74 degrees<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832057169\">\r\n<div data-type=\"problem\" id=\"fs-id1167830705518\">\r\n<p id=\"fs-id1167826987407\">The difference of two supplementary angles is 8 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835301936\">\r\n<div data-type=\"problem\" id=\"fs-id1167835345294\">\r\n<p id=\"fs-id1167834423315\">The difference of two supplementary angles is 88 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832153564\">\r\n<p id=\"fs-id1167834279372\">134 degrees and 46 degrees<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831955188\">\r\n<div data-type=\"problem\" id=\"fs-id1167832128577\">\r\n<p id=\"fs-id1167834309613\">Two angles are supplementary. The measure of the larger angle is four more than three times the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834536390\">\r\n<div data-type=\"problem\" id=\"fs-id1167834536392\">\r\n<p id=\"fs-id1167827957061\">Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167828420936\">\r\n\r\n37 degrees and 143 degrees\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835373672\">\r\n<div data-type=\"problem\" id=\"fs-id1167835349809\">\r\n<p id=\"fs-id1167835349811\">The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826779084\">\r\n<div data-type=\"problem\" id=\"fs-id1167831970020\">\r\n<p id=\"fs-id1167831970023\">The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834219718\">\r\n<p id=\"fs-id1167835512843\">\\(16\\text{\u00b0}\\) and \\(74\\text{\u00b0}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834193531\">\r\n<div data-type=\"problem\" id=\"fs-id1167834193534\">\r\n<p id=\"fs-id1167835622049\">The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826799204\">\r\n<div data-type=\"problem\" id=\"fs-id1167834555360\">\r\n<p id=\"fs-id1167830960829\">The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834524965\">\r\n<p id=\"fs-id1167835263570\">\\(45\\text{\u00b0}\\) and \\(45\\text{\u00b0}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835489374\">\r\n<div data-type=\"problem\" id=\"fs-id1167828411026\">\r\n<p id=\"fs-id1167828411028\">Wayne is hanging a string of lights 45 feet long around the three sides of his patio, which is adjacent to his house. The length of his patio, the side along the house, is five feet longer than twice its width. Find the length and width of the patio.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835335808\">\r\n<div data-type=\"problem\" id=\"fs-id1167835335810\">\r\n<p id=\"fs-id1167835241096\">Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his front yard. The length is five feet less than three times the width. Find the length and width of the fencing.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834377414\">\r\n<p id=\"fs-id1167834377416\">Width is 41 feet and length is 118 feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830865390\">\r\n<div data-type=\"problem\" id=\"fs-id1167831959127\">\r\n<p id=\"fs-id1167831959129\">A frame around a family portrait has a perimeter of 90 inches. The length is fifteen less than twice the width. Find the length and width of the frame.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834194513\">\r\n<div data-type=\"problem\" id=\"fs-id1167835194860\">\r\n<p id=\"fs-id1167835375429\">The perimeter of a toddler play area is 100 feet. The length is ten more than three times the width. Find the length and width of the play area.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835190449\">\r\n<p id=\"fs-id1167831888088\">Width is 10 feet and length is 40 feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167830697840\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\r\n<p id=\"fs-id1167835353585\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167835334297\">\r\n<div data-type=\"problem\" id=\"fs-id1167835479058\">\r\n<p id=\"fs-id1167835173769\">Sarah left Minneapolis heading east on the interstate at a speed of 60 mph. Her sister followed her on the same route, leaving two hours later and driving at a rate of 70 mph. How long will it take for Sarah\u2019s sister to catch up to Sarah?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834330007\">\r\n<div data-type=\"problem\" id=\"fs-id1167835510716\">\r\n<p id=\"fs-id1167835198385\">College roommates John and David were driving home to the same town for the holidays. John drove 55 mph, and David, who left an hour later, drove 60 mph. How long will it take for David to catch up to John?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831985703\">\r\n<p id=\"fs-id1167834539187\">11 hours<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826978939\">\r\n<div data-type=\"problem\" id=\"fs-id1167826978942\">\r\n<p id=\"fs-id1167834120674\">At the end of spring break, Lucy left the beach and drove back towards home, driving at a rate of 40 mph. Lucy\u2019s friend left the beach for home 30 minutes (half an hour) later, and drove 50 mph. How long did it take Lucy\u2019s friend to catch up to Lucy?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835305691\">\r\n<div data-type=\"problem\" id=\"fs-id1167835305693\">\r\n<p id=\"fs-id1167835355663\">Felecia left her home to visit her daughter driving 45 mph. Her husband waited for the dog sitter to arrive and left home twenty minutes (1\/3 hour) later. He drove 55 mph to catch up to Felecia. How long before he reaches her?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834324665\">\r\n<p id=\"fs-id1167834324667\">\\(1.5\\) hour<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835231291\">\r\n<div data-type=\"problem\" id=\"fs-id1167835231293\">\r\n<p id=\"fs-id1167832211867\">The Jones family took a 12-mile canoe ride down the Indian River in two hours. After lunch, the return trip back up the river took three hours. Find the rate of the canoe in still water and the rate of the current.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834593438\">\r\n<div data-type=\"problem\" id=\"fs-id1167834593441\">\r\n<p id=\"fs-id1167834532384\">A motor boat travels 60 miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167831162731\">Boat rate is 16 mph and current rate is 4 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831081632\">\r\n<div data-type=\"problem\" id=\"fs-id1167835373743\">\r\n<p id=\"fs-id1167835373746\">A motor boat traveled 18 miles down a river in two hours but going back upstream, it took \\(4.5\\) hours due to the current. Find the rate of the motor boat in still water and the rate of the current. (Round to the nearest hundredth.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832057264\">\r\n<div data-type=\"problem\" id=\"fs-id1167832057266\">\r\n<p id=\"fs-id1167834533568\">A river cruise boat sailed 80 miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167831191465\">\r\n<p id=\"fs-id1167831837698\">Boat rate is 18 mph and current rate is 2 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831228810\">\r\n<div data-type=\"problem\" id=\"fs-id1167831228812\">\r\n<p id=\"fs-id1167835274915\">A small jet can fly 1072 miles in 4 hours with a tailwind but only 848 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167834120690\">\r\n<div data-type=\"problem\" id=\"fs-id1167834432334\">\r\n<p id=\"fs-id1167834432337\">A small jet can fly 1435 miles in 5 hours with a tailwind but only 1,215 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834094759\">\r\n<p id=\"fs-id1167834094761\">Jet rate is 265 mph and wind speed is 22 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832066764\">\r\n<div data-type=\"problem\" id=\"fs-id1167835498833\">\r\n<p id=\"fs-id1167835498835\">A commercial jet can fly 868 miles in 2 hours with a tailwind but only 792 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167835312119\">\r\n<div data-type=\"problem\" id=\"fs-id1167835312121\">\r\n<p id=\"fs-id1167830865979\">A commercial jet can fly 1,320 miles in 3 hours with a tailwind but only 1170 miles in 3 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167835306122\">\r\n<p id=\"fs-id1167835226254\">Jet rate is 415 mph and wind speed is 25 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835321214\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167835283060\">\r\n<div data-type=\"problem\" id=\"fs-id1167835283062\">\r\n<p id=\"fs-id1167834161535\">Write an application problem similar to <a href=\"#fs-id1167834556567\" class=\"autogenerated-content\">(Figure)<\/a>. Then translate to a system of equations and solve it.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167831912147\">\r\n<div data-type=\"problem\" id=\"fs-id1167831912149\">\r\n<p id=\"fs-id1167831106889\">Write a uniform motion problem similar to <a href=\"#fs-id1167835186838\" class=\"autogenerated-content\">(Figure)<\/a> that relates to where you live with your friends or family members. Then translate to a system of equations and solve it.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167834190356\">\r\n<p id=\"fs-id1167835230726\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835311989\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167834489878\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167834324745\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve direct translation applications, solve geometry applications, solve uniform motion applications. The remaining columns are empty.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve direct translation applications, solve geometry applications, solve uniform motion applications. The remaining columns are empty.\" \/><\/span>\r\n<p id=\"fs-id1167828426742\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl id=\"fs-id1167834183572\">\r\n \t<dt>complementary angles<\/dt>\r\n \t<dd id=\"fs-id1167832010323\">Two angles are complementary if the sum of the measures of their angles is 90 degrees.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167835623490\">\r\n \t<dt>supplementary angles<\/dt>\r\n \t<dd>Two angles are supplementary if the sum of the measures of their angles is 180 degrees.<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Solve direct translation applications<\/li>\n<li>Solve geometry applications<\/li>\n<li>Solve uniform motion applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830703077\" class=\"be-prepared\">\n<p id=\"fs-id1167830698775\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835341827\" type=\"1\">\n<li>The sum of twice a number and nine is 31. Find the number.\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766#fs-id1167836525150\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Twins Jon and Ron together earned ?96,000 last year. Ron earned ?8000 more than three times what Jon earned. How much did each of the twins earn?\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766#fs-id1167836606849\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains.\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2#fs-id1167836309166\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Solve Direct Translation Applications<\/h3>\n<p id=\"fs-id1167835302397\">Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we\u2019ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.<\/p>\n<div data-type=\"note\" id=\"fs-id1167828435685\" class=\"howto\">\n<div data-type=\"title\">Solve applications with systems of equations.<\/div>\n<ol id=\"fs-id1167834120308\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167830865965\">We solved number problems with one variable earlier. Let\u2019s see how differently it works using two variables.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834556567\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831954242\">\n<div data-type=\"problem\" id=\"fs-id1167835174952\">\n<p id=\"fs-id1167830960764\">The sum of two numbers is zero. One number is nine less than the other. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835364658\">\n<table id=\"fs-id1167832010472\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for 2 numbers. Step 3 is to name the two numbers n and m. Step 4 is to translate into a system of equations. We get n plus m equal to 0 and n equal to m minus 9. Step 5 is to solve the system of equations. We substitute m minus n for n in the first equation and solve for m. We get m equal to 9 by 2. We substitute this in the second equation. Solving for n, we get n equal to minus 9 by 2. Step 6 is to check if the answer make sense in the problem. Step 7 is to answer the problem. The numbers are 9 by 2 and minus 9 by 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for two numbers.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-602c4d5dfa06ccbd52974998a84a8f4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"29\" style=\"vertical-align: 0px;\" \/> the first number.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34e51f0734df81122263d12cbeebe466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"33\" style=\"vertical-align: 0px;\" \/> the second number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The sum of two numbers is zero.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835216620\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">One number is nine less than the other.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834395672\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832031159\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equations. We will use substitution<\/p>\n<div data-type=\"newline\"><\/div>\n<p>since the second equation is solved<\/p>\n<div data-type=\"newline\"><\/div>\n<p>for <em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">m<\/em> \u2212 9 for <em data-effect=\"italics\">n<\/em> in the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834094607\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">m<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826977660\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535269\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834525104\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16bd110d9406c389f9c9a98bdd3ae2df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> into the second equation<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and then solve for <em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835311214\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835201041\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835321418\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Do these numbers make sense in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the problem? We will leave this to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>you!<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The numbers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a0158b653269485337be6797772c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0460e95f6f820261fade7dd5882d8605_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835197823\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835634616\">\n<div data-type=\"problem\" id=\"fs-id1167835254462\">\n<p id=\"fs-id1167834228969\">The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835174756\">\n<p id=\"fs-id1167835174892\">3, 7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826994044\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835390449\">\n<div data-type=\"problem\" id=\"fs-id1167835254405\">\n<p id=\"fs-id1167834515363\">The sum of two numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e45e9e13a8901226d7601cbfa15df84a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> One number is 10 less than the other. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378190\">\n<p>2, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835186838\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167830770008\">\n<div data-type=\"problem\" id=\"fs-id1167831872425\">\n<p>Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her ?25,000 plus ?15 for each training session. Option B would pay her <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1528ca5a8527815ad8dcb731f8da1ac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#48;&#44;&#48;&#48;&#48;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> for each training session. How many training sessions would make the salary options equal?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167826994030\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the number of training sessions that would make the pay equal. Step 3 is to name what we are looking for. We name Heather\u2019s salary s and the number of training sessions n. Step 4 is to translate into a system of equations. Option A would pay her 25,000 dollars plus 15 dollars for each training session. So, s is 25,000 plus 15n. Option B would pay her 10,000 dollars plus 40 dollars for each training session. So, s is 10,000 plus 40n. Step 5 is to solve the system of equations. We substitute 25,000 plus 15n for s in the second equation and solve for n. We get n equal to 600. Step 6 is to check if the answer make sense in the problem. Are 600 training sessions a year reasonable? Are the two options equal when n equal to 600? Step 7 is to answer the problem. The salary options would be equal for 600 training sessions.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the number of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>training sessions that would make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the pay equal.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12ac9f5dee0c0d9acdb247ec52211d05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> Heather\u2019s salary.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee1750ab3f138676ed789405f7cec8c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> the number of training sessions<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Option A would pay her ?25,000<\/p>\n<div data-type=\"newline\"><\/div>\n<p>plus ?15 for each training<\/p>\n<div data-type=\"newline\"><\/div>\n<p>session.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835165797\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Option B would pay her ?10,000<\/p>\n<div data-type=\"newline\"><\/div>\n<p>+ ?40 for each training session.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792546807\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is shown.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826941234\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>We will use substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835332871\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute 25,000 +15<em data-effect=\"italics\">n<\/em> for <em data-effect=\"italics\">s<\/em> in the second<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305413\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826799100\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Are 600 training sessions a year reasonable?<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Are the two options equal when <em data-effect=\"italics\">n<\/em> = 600?<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The salary options would be equal for 600 training<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sessions.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835304408\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835366813\">\n<div data-type=\"problem\" id=\"fs-id1167835545400\">\n<p id=\"fs-id1167835335775\">Geraldine has been offered positions by two insurance companies. The first company pays a salary of ?12,000 plus a commission of ?100 for each policy sold. The second pays a salary of ?20,000 plus a commission of ?50 for each policy sold. How many policies would need to be sold to make the total pay the same?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830868501\">\n<p id=\"fs-id1167832068503\">160 policies<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835511221\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835303466\">\n<div data-type=\"problem\" id=\"fs-id1167835187907\">\n<p id=\"fs-id1167835237602\">Kenneth currently sells suits for company A at a salary of ?22,000 plus a ?10 commission for each suit sold. Company B offers him a position with a salary of ?28,000 plus a ?4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835320702\">1000 suits<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168755200519\">As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834064452\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834464405\">\n<p id=\"fs-id1167831882418\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835231147\">When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for 20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes on the elliptical trainer and 30 minutes circuit training she burned 473 calories. How many calories does she burn for each minute on the elliptical trainer? How many calories for each minute of circuit training?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835320319\">\n<table id=\"fs-id1167835327644\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the number of calories burned each minute on the elliptical trainer and each minute of circuit training. Step 3 is to name what we are looking for. Let e be the number of calories burned per minute on the elliptical trainer and c be the number of calories burned per minute of circuit training. Step 4 is to translate into a system of equations. 10e plus 20c is 278 and 20e plus 30c is 473. Step 5 is to solve the system of equations. We multiply the first by minus 2 to get opposite coefficients of e. Simplify and add the equations and solve for c. We get c equal to 8.3. Substituting this value in either of the original equations, we get e equal to 11.2. Step 6 is to check the answer in the problem. Step 7 is to answer the problem. Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the number of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>calories burned each minute on the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>elliptical trainer and each minute of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>circuit training.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92345ce99bdfb0ef76d06a5d552be726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> number of calories burned per<\/p>\n<div data-type=\"newline\"><\/div>\n<p>minute on the elliptical trainer.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9979648413b4c8f3eb6fceacfa83e6c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"25\" style=\"vertical-align: 0px;\" \/> number of calories burned per<\/p>\n<div data-type=\"newline\"><\/div>\n<p>minute while circuit training<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">10 minutes on the elliptical and circuit<\/p>\n<div data-type=\"newline\"><\/div>\n<p>training for 20 minutes, burned<\/p>\n<div data-type=\"newline\"><\/div>\n<p>278 calories<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305222\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">20 minutes on the elliptical and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>30 minutes of circuit training burned<\/p>\n<div data-type=\"newline\"><\/div>\n<p>473 calories<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053714\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the first equation by \u22122 to get<\/p>\n<div data-type=\"newline\"><\/div>\n<p>opposite coefficients of <em data-effect=\"italics\">e<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832010294\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify and add the equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Solve for <em data-effect=\"italics\">c<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237274\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">c<\/em> = 8.3 into one of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>original equations to solve for <em data-effect=\"italics\">e<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831117433\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Check the math on your own.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509656\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Jenna burns 8.3 calories per minute<\/p>\n<div data-type=\"newline\"><\/div>\n<p>circuit training and 11.2 calories per<\/p>\n<div data-type=\"newline\"><\/div>\n<p>minute while on the elliptical trainer.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834501534\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167830964293\">\n<p>Translate to a system of equations and then solve:<\/p>\n<p>Mark went to the gym and did 40 minutes of Bikram hot yoga and 10 minutes of jumping jacks. He burned 510 calories. The next time he went to the gym, he did 30 minutes of Bikram hot yoga and 20 minutes of jumping jacks burning 470 calories. How many calories were burned for each minute of yoga? How many calories were burned for each minute of jumping jacks?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834064368\">Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831949038\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834346518\">\n<div data-type=\"problem\" id=\"fs-id1167835414903\">\n<p id=\"fs-id1167834539344\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835410359\">Erin spent 30 minutes on the rowing machine and 20 minutes lifting weights at the gym and burned 430 calories. During her next visit to the gym she spent 50 minutes on the rowing machine and 10 minutes lifting weights and burned 600 calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835334955\">\n<p>Erin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832152518\">\n<h3 data-type=\"title\">Solve Geometry Applications<\/h3>\n<p id=\"fs-id1167835241391\">We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.<\/p>\n<p id=\"fs-id1167830702417\">The measures of two <span data-type=\"term\">complementary angles<\/span> add to 90 degrees. The measures of two <span data-type=\"term\">supplementary angles<\/span> add to 180 degrees.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Complementary and Supplementary Angles<\/div>\n<p id=\"fs-id1167835514605\">Two angles are complementary if the sum of the measures of their angles is 90 degrees.<\/p>\n<p id=\"fs-id1167835329300\">Two angles are supplementary if the sum of the measures of their angles is 180 degrees.<\/p>\n<\/div>\n<p id=\"fs-id1167835200476\">If two angles are complementary, we say that <em data-effect=\"italics\">one angle is the complement of the other.<\/em><\/p>\n<p id=\"fs-id1167835201883\">If two angles are supplementary, we say that <em data-effect=\"italics\">one angle is the supplement of the other.<\/em><\/p>\n<div data-type=\"example\" id=\"fs-id1167835363488\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831228660\">\n<p id=\"fs-id1167832153433\">Translate to a system of equations and then solve.<\/p>\n<p id=\"fs-id1167834059688\">The difference of two complementary angles is 26 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281517\">\n<p id=\"fs-id1167835371347\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae8019c50c385f9ccbfe90095fe012eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#119;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#109;&#101;&#97;&#115;&#117;&#114;&#101;&#32;&#111;&#102;&#32;&#101;&#97;&#99;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#103;&#108;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#119;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#109;&#101;&#97;&#115;&#117;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#97;&#110;&#103;&#108;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#8195;&#8195;&#121;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#109;&#101;&#97;&#115;&#117;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#97;&#110;&#103;&#108;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#111;&#32;&#97;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#97;&#110;&#103;&#108;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#109;&#101;&#110;&#116;&#97;&#114;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#9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title=\"Rendered by QuickLaTeX.com\" height=\"732\" width=\"717\" style=\"vertical-align: -361px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831985764\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834195982\">\n<p id=\"fs-id1167835544861\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167834192154\">The difference of two complementary angles is 20 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831922064\">The angle measures are 55 and 35.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835365592\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835346376\">\n<div data-type=\"problem\" id=\"fs-id1167835355260\">\n<p id=\"fs-id1167835419750\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835330289\">The difference of two complementary angles is 80 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835390037\">\n<p>The angle measures are 5 and 85.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168756438299\">In the next example, we remember that the measures of supplementary angles add to 180.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835342807\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831239714\">\n<div data-type=\"problem\" id=\"fs-id1167835360777\">\n<p id=\"fs-id1167830963487\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167826781076\">Two angles are supplementary. The measure of the larger angle is twelve degrees less than five times the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834194375\">\n<table id=\"fs-id1167835325851\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the measure of each angle. Step 3 is to name what we are looking for. Let x be the measure of the first angle and y be the measure of the second angle. Step 4 is to translate into a system of equations. The angles are supplementary. So, x plus y is 180 and y is 5 x minus 12. Step 5 is to solve the system of equations by substitution. Substitute 5 x minus 12 for y in the first equation. Solving for x, we get x equal to 32. Substitute this in the second equation and solve for y. We get y equal to 148. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The angle measures are 148 and 32.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for measure of each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>angle.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> the measure of the first angle.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-300a486b79d88c3b7696ef4b8978050b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\" \/> the measure of the second angle<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The angles are supplementary.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064608\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The larger angle is twelve less than five<\/p>\n<div data-type=\"newline\"><\/div>\n<p>times the smaller angle.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834097914\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is shown:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831112459\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute 5<em data-effect=\"italics\">x<\/em> \u2212 12 for <em data-effect=\"italics\">y<\/em> in the first equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390372\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Substitute 32 for <em data-effect=\"italics\">x<\/em> in the second<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation, then solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336028\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167830991319\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834394937\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The angle measures are 148 and 32 degrees.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835217900\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831887926\">\n<div data-type=\"problem\" id=\"fs-id1167830960497\">\n<p id=\"fs-id1167835335238\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167826783686\">Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420011\">\n<p id=\"fs-id1167835344064\">The angle measures are 42 and 138.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826778516\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835390453\">\n<p id=\"fs-id1167834429847\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835329416\">Two angles are supplementary. The measure of the larger angle is 18 less than twice the measure of the smaller angle. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834184127\">\n<p id=\"fs-id1167831970277\">The angle measures are 66 and 114.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835280307\">Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835167148\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835367895\">\n<div data-type=\"problem\" id=\"fs-id1167835384327\">\n<p>4.19<\/p>\n<p id=\"fs-id1167831881234\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831822903\">\n<p id=\"fs-id1167834583384\">We will draw and label a figure.<\/p>\n<table id=\"fs-id1167831884545\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the measures of the two angles of the triangle. Step 3 is to name what we are looking for. Let A equal the measure of the first angle. Let B equal the measure of the second angle. Step 4 is to translate into a system of equations. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. This means that A equals 3 B plus 10. The sum of the measures of the angles of a triangle is 180. A plus B plus 90 equals 180. The system is A equals 3 B plus 10 and A plus B plus 90 equals 180. Step 5 is to solve the system of equations. We will use substitution since the first equation is solved for A. Because A equals 3 B plus 10, we substitute 3 B plus 10 in for A in the second equation. This gives us 3 B plus 10, in parentheses, plus B plus 90 equals 180. Solve for B. 4 B plus 100 equals 180. 4 B equals 80. B equals 20. Substitute B equals 20 into the first equation and then solve for A. A equals 3 times 20 plus 10. A equals 70. Step 6 is to check the answer in the problem. We will leave this to you. Step 7 is to answer the question. The measures of the small angles are 20 and 70.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832065615\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the measures of the angles.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fdf9894e43ac86e8b63ba96c72e7181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"27\" style=\"vertical-align: 0px;\" \/> the measure of the first angle.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdf71bbe58eb1a3f8789c2cc199d76dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: 0px;\" \/> the measure of the second angle<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The sum of the measures of the angles of a triangle is 180.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835279918\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is shown.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835337274\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the system of equations. We will use substitution since the first equation is solved for <em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835640016\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8e1704eb670734bb89d2a7967d71996_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#98;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -2px;\" \/> for <em data-effect=\"italics\">a<\/em> in the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835202914\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">b<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254030\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c442e950107efbd2057bfc4f551216eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> into the first equation and then solve for <em data-effect=\"italics\">a.<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834063905\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We will leave this to you!<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The measures of the small angles are 20 and 70 degrees.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834066299\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834402920\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831958101\">The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830705915\">\n<p id=\"fs-id1167834403373\">22, 68<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834469592\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834194118\">\n<div data-type=\"problem\" id=\"fs-id1167826874759\">\n<p id=\"fs-id1167834228931\">The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826802370\">\n<p id=\"fs-id1167834532312\">36, 54<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168756447619\">Often it is helpful when solving geometry applications to draw a picture to visualize the situation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835231793\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835254341\">\n<div data-type=\"problem\" id=\"fs-id1167835306773\">\n<p id=\"fs-id1167835380024\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835483610\">Randall has 125 feet of fencing to enclose the part of his backyard adjacent to his house. He will only need to fence around three sides, because the fourth side will be the wall of the house. He wants the length of the fenced yard (parallel to the house wall) to be 5 feet more than four times as long as the width. Find the length and the width.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064239\">\n<table id=\"fs-id1167832054759\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. Step 2 is to identify what we are looking for. We are looking for the length and width. Step 3 is to name what we are looking for. Let L be the length and W be the width. Step 4 is to translate into a system of equations. L plus 2W is 125 and L is 4W plus 5. Step 5 is to solve the system of equations. Substituting the second equation into the first and solving for W, we get W equal to 20. Substituting this into the second equation, we get L equal to 85. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The length is 85 feet and width is 20 feet.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the length and width.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237826\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-549284767161ef0d3728f1f8c7bce365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/> the length of the fenced yard.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7b6466a27284f30601bb2982e82850c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#87;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"37\" style=\"vertical-align: 0px;\" \/> the width of the fenced yard<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into a system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">One lenth and two widths equal 125.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834395872\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The length will be 5 feet more than<\/p>\n<div data-type=\"newline\"><\/div>\n<p>four times the width.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The system is shown.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\">Step 5. Solve<\/strong> The system of equations<\/p>\n<div data-type=\"newline\"><\/div>\n<p>by substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826994454\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">L<\/em> = 4<em data-effect=\"italics\">W<\/em> + 5 into the first<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation, then solve for <em data-effect=\"italics\">W<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358000\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345478\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute 20 for <em data-effect=\"italics\">W<\/em> in the second<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation, then solve for <em data-effect=\"italics\">L<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834598458\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The length is 85 feet and the width is 20 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830697622\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834191195\">\n<div data-type=\"problem\" id=\"fs-id1167834300837\">\n<p id=\"fs-id1167831921373\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167832066931\">Mario wants to put a fence around the pool in his backyard. Since one side is adjacent to the house, he will only need to fence three sides. There are two long sides and the one shorter side is parallel to the house. He needs 155 feet of fencing to enclose the pool. The length of the long side is 10 feet less than twice the width. Find the length and width of the pool area to be enclosed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835367103\">\n<p id=\"fs-id1167835498160\">The length is 60 feet and the width is 35 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832066759\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831881110\">\n<div data-type=\"problem\" id=\"fs-id1167835300576\">\n<p id=\"fs-id1167828433171\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167834151997\">Alexis wants to build a rectangular dog run in her yard adjacent to her neighbor\u2019s fence. She will use 136 feet of fencing to completely enclose the rectangular dog run. The length of the dog run along the neighbor\u2019s fence will be 16 feet less than twice the width. Find the length and width of the dog run.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835368759\">\n<p id=\"fs-id1167832058855\">The length is 60 feet and the width is 38 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835180813\">\n<h3 data-type=\"title\">Solve uniform motion applications<\/h3>\n<p id=\"fs-id1167828421536\">We used a table to organize the information in uniform motion problems when we introduced them earlier. We\u2019ll continue using the table here. The basic equation was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-164fcc881021ce79be78a8adbd303299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> where <em data-effect=\"italics\">D<\/em> is the distance traveled, <em data-effect=\"italics\">r<\/em> is the rate, and <em data-effect=\"italics\">t<\/em> is the time.<\/p>\n<p id=\"fs-id1167834279160\">Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831920196\">\n<div data-type=\"problem\" id=\"fs-id1167835304306\">\n<p id=\"fs-id1167835351018\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167828240732\">Joni left St. Louis on the interstate, driving west towards Denver at a speed of 65 miles per hour. Half an hour later, Kelly left St. Louis on the same route as Joni, driving 78 miles per hour. How long will it take Kelly to catch up to Joni?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835261754\">\n<p id=\"fs-id1167834229330\">A diagram is useful in helping us visualize the situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834147077\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1167830700663\"><strong data-effect=\"bold\">Identify and name<\/strong> what we are looking for. A chart will help us organize the data. We know the rates of both Joni and Kelly, and so we enter them in the chart. We are looking for the length of time Kelly, <em data-effect=\"italics\">k<\/em>, and Joni, <em data-effect=\"italics\">j<\/em>, will each drive.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167826996003\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1167834189936\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bee03565ba220379247c30f1ee89da9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&middot;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> we can fill in the Distance column.<\/p>\n<p id=\"fs-id1167835513925\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/p>\n<p id=\"fs-id1167835308664\">To make the system of equations, we must recognize that Kelly and Joni will drive the same distance. So,<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147949384\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72a8b95e8c4a9a2929983f31fa2e9bae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#53;&#106;&#61;&#55;&#56;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"76\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167834473791\">Also, since Kelly left later, her time will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> hour less than Joni\u2019s time. So,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c7bb3e6ae35e1d1cdccccbc6cfbbd19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#106;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#119;&#32;&#119;&#101;&#32;&#104;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#106;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#53;&#106;&#61;&#55;&#56;&#107;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#32;&#111;&#102;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#98;&#121;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#106;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#106;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#53;&#106;&#61;&#55;&#56;&#107;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#53;&#106;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#106;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#53;&#106;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#56;&#106;&#45;&#51;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#51;&#106;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#106;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#75;&#101;&#108;&#108;&#121;&#39;&#115;&#32;&#116;&#105;&#109;&#101;&#44;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#106;&#61;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#44;&#32;&#116;&#104;&#101;&#110;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#106;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#111;&#110;&#105;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#49;&#57;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#108;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#101;&#108;&#108;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#57;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#108;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#101;&#115;&#44;&#32;&#116;&#104;&#101;&#121;&#32;&#119;&#105;&#108;&#108;&#32;&#104;&#97;&#118;&#101;&#32;&#116;&#114;&#97;&#118;&#101;&#108;&#101;&#100;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#121;&#32;&#109;&#101;&#101;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#110;&#115;&#119;&#101;&#114;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#101;&#108;&#108;&#121;&#32;&#119;&#105;&#108;&#108;&#32;&#99;&#97;&#116;&#99;&#104;&#32;&#117;&#112;&#32;&#116;&#111;&#32;&#74;&#111;&#110;&#105;&#32;&#105;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#46;&#32;&#66;&#121;&#32;&#116;&#104;&#101;&#110;&#44;&#32;&#74;&#111;&#110;&#105;&#32;&#119;&#105;&#108;&#108;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#97;&#118;&#101;&#32;&#116;&#114;&#97;&#118;&#101;&#108;&#101;&#100;&#32;&#51;&#32;&#104;&#111;&#117;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831970009\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832052250\">\n<p id=\"fs-id1167834186157\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167834059465\">Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834053473\">\n<p id=\"fs-id1167834111767\">It will take Clark 4 hours to catch Mitchell.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831883648\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834396468\">\n<div data-type=\"problem\" id=\"fs-id1167835533572\">\n<p id=\"fs-id1167834432254\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167831880362\">Charlie left his mother\u2019s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-721b129babdfd953be5b51c57ba9599f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/> later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835320770\">\n<p id=\"fs-id1167826783003\">It will take Sally <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bb20d9986d47c316031bc35dc35a37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"17\" style=\"vertical-align: -6px;\" \/> hours to catch up to Charlie.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835306358\">Many real-world applications of uniform motion arise because of the effects of currents\u2014of water or air\u2014on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.<\/p>\n<p id=\"fs-id1167835191090\">Let\u2019s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.<\/p>\n<p id=\"fs-id1167834472600\">The images below show how a river current affects the speed at which a boat is actually travelling. We\u2019ll call the speed of the boat in still water <em data-effect=\"italics\">b<\/em> and the speed of the river current <em data-effect=\"italics\">c<\/em>.<\/p>\n<p id=\"fs-id1167834098208\">The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat\u2019s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c26e81df526218cdff7c2b8222d266_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167827958519\" data-alt=\"Figure shows a boat and two horizontal arrows, both pointing left. The one to the left of the boat is b and the one to the right is c.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a boat and two horizontal arrows, both pointing left. The one to the left of the boat is b and the one to the right is c.\" \/><\/span><\/p>\n<p id=\"fs-id1167835181916\">Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat\u2019s actual speed is slower than its speed in still water. The actual speed of the boat is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11c6adb5100efd0caff58f74f8683bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#45;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835610140\" data-alt=\"Figure shows a boat and two horizontal arrows to its left. One, labeled b, points left and the other, labeled c, points right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a boat and two horizontal arrows to its left. One, labeled b, points left and the other, labeled c, points right.\" \/><\/span><\/p>\n<p id=\"fs-id1167835333178\">We\u2019ll put some numbers to this situation in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835534188\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835238926\">\n<div data-type=\"problem\" id=\"fs-id1167834053490\">\n<p id=\"fs-id1167835365522\">Translate to a system of equations and then solve.<\/p>\n<p id=\"fs-id1167835356872\">A river cruise ship sailed 60 miles downstream for 4 hours and then took 5 hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831883744\">\n<table id=\"fs-id1167835343141\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. This is a uniform motion problem and a picture will help us visualize the situation. The picture shows a wavy line pointing right. This is labeled c. A straight arrow pointing right is labeled 4 hours. Another arrow pointing left is labeled 5 hours. The length of arrows is labeled 60 miles. Step 2 is to identify what we are looking for. We are looking for the speed of the ship in still water and the speed of the current. Step 3 is to name what we are looking for. Let s be the rate of ship in still water and c be the rate of current. The ship goes downstream and then upstream. Going downstream, the current helps the ship and so the ship's actual rate is s plus c. Going upstream, the current slows the ship and so the actual rate is s minus c. Step 4 is to translate into a system of equations. 4 open parentheses s plus c close parentheses is 60 and 5 open parentheses s minus c close parentheses is 60. Step 5 is to solve the system of equations. Multiply the top equation by 5 and the bottom one by 4. By adding them and solving for s, we get s equal to 13.5. By substituting, we get c equal to 1.5. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The rate of ship is 13.5 and rate of current is 1.5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\">This is a uniform motion problem and a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>picture will help us visualize the situation.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835341268\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the speed of the ship<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in still water and the speed of the current.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12ac9f5dee0c0d9acdb247ec52211d05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> the rate of the ship in still water.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-781c84447f692b5c948028970a260f7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"25\" style=\"vertical-align: 0px;\" \/> the rate of the current<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">A chart will help us organize the information.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The ship goes downstream and then upstream.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Going downstream, the current helps the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>ship and so the ship&#8217;s actual rate is <em data-effect=\"italics\">s<\/em> + <em data-effect=\"italics\">c<\/em>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Going upstream, the current slows the ship<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and so the actual rate is <em data-effect=\"italics\">s<\/em> \u2212 <em data-effect=\"italics\">c<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556600\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Downstream it takes 4 hours.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Upstream it takes 5 hours.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Each way the distance is 60 miles.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since rate times time is distance, we can<\/p>\n<div data-type=\"newline\"><\/div>\n<p>write the system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the system of equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Distribute to put both equations in standard<\/p>\n<div data-type=\"newline\"><\/div>\n<p>form, then solve by elimination.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327540\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the top equation by 5 and the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>bottom equation by 4.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Add the equations, then solve for <em data-effect=\"italics\">s<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835361923\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">s<\/em> = 13.5 into of the original<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835512731\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Check<\/strong> the answer in the problem.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The downstream rate would be<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e463b720bba533974d589ffecb92524f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#46;&#53;&#43;&#49;&#46;&#53;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"115\" style=\"vertical-align: -2px;\" \/> mph.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In 4 hours the ship would travel<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0435ffdcf2591565da1de91d971eacb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&middot;&#52;&#61;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/> miles.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The upstream rate would be<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d27b2cc859282db5e2666496b9e4e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#46;&#53;&#45;&#49;&#46;&#53;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"115\" style=\"vertical-align: -1px;\" \/> mph.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In 5 hours the ship would travel<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afeab044af03f8cba4ab40874b8b6cc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&middot;&#53;&#61;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/> miles.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The rate of the ship is 13.5 mph and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the rate of the current is 1.5 mph.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834099440\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834183359\">\n<div data-type=\"problem\" id=\"fs-id1167831832521\">\n<p id=\"fs-id1167834183093\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167831955755\">A Mississippi river boat cruise sailed 120 miles upstream for 12 hours and then took 10 hours to return to the dock. Find the speed of the river boat in still water and the speed of the river current.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834227436\">\n<p id=\"fs-id1167834367202\">The rate of the boat is 11 mph and the rate of the current is 1 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834314564\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834279563\">\n<div data-type=\"problem\" id=\"fs-id1167835349130\">\n<p id=\"fs-id1167830909024\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835284768\">Jason paddled his canoe 24 miles upstream for 4 hours. It took him 3 hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826997660\">\n<p id=\"fs-id1167835352364\">The speed of the canoe is 7 mph and the speed of the current is 1 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834432815\">Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We\u2019ll see this in the next example. A wind current in the same direction as the plane is flying is called a <em data-effect=\"italics\">tailwind<\/em>. A wind current blowing against the direction of the plane is called a <em data-effect=\"italics\">headwind<\/em>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831836495\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832015630\">\n<div data-type=\"problem\" id=\"fs-id1167834284501\">\n<p id=\"fs-id1167835338954\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167835327022\">A private jet can fly 1,095 miles in three hours with a tailwind but only 987 miles in three hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319175\">\n<table id=\"fs-id1167834327609\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to read the problem. This is a uniform motion problem and a picture will help us visualize the situation. The picture shows a wavy line pointing right. This is labeled wind, 3 hours. A straight arrow pointing right is labeled j plus w equals 365, 1095 miles. Another arrow, pointing left is labeled j minus w equals 329, 987 miles. Step 2 is to identify what we are looking for. We are looking for the speed of the jet in still air and the speed of the wind. Step 3 is to name what we are looking for. Let j be the speed of jet in still air and w be the speed of wind. Step 4 is to translate into a system of equations. 3j plus 3w is 1095 and 3j minus 3w is 987. Step 5 is to solve the system of equations. Solving by elimination, we get j equal to 347. Solving by substitution we get w equal to 18. Step 6 is to check the answer in the problem. Step 7 is to answer the question. The rate of the jet is 347 mph and rate of wind is 18 mph.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\">This is a uniform motion problem and a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>picture will help us visualize.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327739\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the speed of the jet<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in still air and the speed of the wind.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77f99e21c7f0321e0792321c2922f946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: -4px;\" \/> the speed of the jet in still air.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da4a9411034124dca7acc84c76b9cabf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#119;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"31\" style=\"vertical-align: 0px;\" \/> the speed of the wind.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">A chart will help us organize the information.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The jet makes two trips\u2014one in a tailwind<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and one in a headwind.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In a tailwind, the wind helps the jet and so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the rate is <em data-effect=\"italics\">j<\/em> + <em data-effect=\"italics\">w<\/em>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In a headwind, the wind slows the jet and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>so the rate is <em data-effect=\"italics\">j<\/em> \u2012 <em data-effect=\"italics\">w<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834397046\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Each trip takes 3 hours.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In a tailwind the jet flies 1,095 miles.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In a headwind the jet flies 987 miles.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since rate times time is distance, we get the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>system of equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826941282\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the system of equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Distribute, then solve by elimination.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Add, and solve for <em data-effect=\"italics\">j<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831890692\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">j<\/em> = 347 into one of the original<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equations, then solve for <em data-effect=\"italics\">w<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835341957\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Check<\/strong> the answer in the problem.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>With the tailwind, the actual rate of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>jet would be<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fdd2ad3fc114821b40c81fe9066084c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#52;&#55;&#43;&#49;&#56;&#61;&#51;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"116\" style=\"vertical-align: -2px;\" \/> mph.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In 3 hours the jet would travel<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-562712d715499f3ffb21654e4f97c2e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#53;&middot;&#51;&#61;&#49;&#44;&#48;&#57;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -4px;\" \/> miles<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Going into the headwind, the jet\u2019s actual<\/p>\n<div data-type=\"newline\"><\/div>\n<p>rate would be<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bf50ee39e94bc5e7bbe0cabeddbef42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#52;&#55;&#45;&#49;&#56;&#61;&#51;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"117\" style=\"vertical-align: -1px;\" \/> mph.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>In 3 hours the jet would travel<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a36cfaae8051deea9fe2c4b886f6553c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#57;&middot;&#51;&#61;&#57;&#56;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"86\" style=\"vertical-align: 0px;\" \/> miles.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The rate of the jet is 347 mph and the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>rate of the wind is 18 mph.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835233774\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835360362\">\n<div data-type=\"problem\" id=\"fs-id1167835368111\">\n<p id=\"fs-id1167834556843\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167834226271\">A small jet can fly 1,325 miles in 5 hours with a tailwind but only 1,035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345606\">\n<p id=\"fs-id1167835234036\">The speed of the jet is 235 mph and the speed of the wind is 30 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826998350\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828435014\">\n<div data-type=\"problem\" id=\"fs-id1167834345458\">\n<p id=\"fs-id1167826977977\">Translate to a system of equations and then solve:<\/p>\n<p id=\"fs-id1167834462714\">A commercial jet can fly 1,728 miles in 4 hours with a tailwind but only 1,536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835322254\">\n<p id=\"fs-id1167835231016\">The speed of the jet is 408 mph and the speed of the wind is 24 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"media-2\">\n<p id=\"fs-id1167832151228\">Access this online resource for additional instruction and practice with systems of equations.<\/p>\n<ul id=\"fs-id1169148234280\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37sysequations\">Systems of Equations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831921394\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167831881094\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How To Solve Applications with Systems of Equations<\/strong>\n<ol id=\"fs-id1167835369775\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose variables to represent those quantities.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into a system of equations.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the system of equations using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835364750\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834141990\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167835207310\"><strong data-effect=\"bold\">Direct Translation Applications<\/strong><\/p>\n<p id=\"fs-id1167835344124\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834535347\">\n<div data-type=\"problem\" id=\"fs-id1167826813970\">\n<p id=\"fs-id1167835348193\">The sum of two number is 15. One number is 3 less than the other. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832055884\">\n<p id=\"fs-id1167835534165\">The sum of two number is 30. One number is 4 less than the other. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835434156\">\n<p>13 and 17<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057971\">\n<div data-type=\"problem\" id=\"fs-id1167832056216\">\n<p id=\"fs-id1167831910371\">The sum of two number is \u221216. One number is 20 less than the other. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395267\">\n<div data-type=\"problem\" id=\"fs-id1167832068556\">\n<p id=\"fs-id1167831880227\">The sum of two number is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39166e0aad54e2681da9cc30b2a3f71e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/> One number is 12 less than the other. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835361560\">\n<p id=\"fs-id1167835253917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e22bf27371ae8e82f2cf9857cc59c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834527974\">\n<div data-type=\"problem\" id=\"fs-id1167835339583\">\n<p id=\"fs-id1167831919812\">The sum of two numbers is 65. Their difference is 25. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835382125\">\n<div data-type=\"problem\" id=\"fs-id1167830696690\">\n<p id=\"fs-id1167835518468\">The sum of two numbers is 37. Their difference is 9. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835342748\">\n<p id=\"fs-id1167835378005\">14 and 23<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834319791\">\n<div data-type=\"problem\" id=\"fs-id1167834053763\">\n<p id=\"fs-id1167835310165\">The sum of two numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcba93e60a649c5e366bcaa6bd08ef02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/> Their difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-696a8a4c53f53bdc0dfb9e6c47fe2589_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/> Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835350540\">\n<div data-type=\"problem\" id=\"fs-id1167834138030\">\n<p id=\"fs-id1167835349416\">The sum of two numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5338429c8c928f42962d7767856343a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"35\" style=\"vertical-align: -1px;\" \/> Their difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5349fb5e458e62a65630d72de575ec7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/> Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835421990\">\n<p id=\"fs-id1167835420760\">22 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fa8b402f683fe9de7dd7bde82ca284f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831824722\">\n<div data-type=\"problem\" id=\"fs-id1167835483590\">\n<p id=\"fs-id1167835381769\">Maxim has been offered positions by two car companies. The first company pays a salary of ?10,000 plus a commission of ?1000 for each car sold. The second pays a salary of ?20,000 plus a commission of ?500 for each car sold. How many cars would need to be sold to make the total pay the same?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834473411\">\n<div data-type=\"problem\" id=\"fs-id1167834516218\">\n<p id=\"fs-id1167831162894\">Jackie has been offered positions by two cable companies. The first company pays a salary of ?14,000 plus a commission of ?100 for each cable package sold. The second pays a salary of ?20,000 plus a commission of ?25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066666\">\n<p>Eighty cable packages would need to be sold to make the total pay the same.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834238729\">\n<div data-type=\"problem\" id=\"fs-id1167835628567\">\n<p id=\"fs-id1167835609615\">Amara currently sells televisions for company A at a salary of ?17,000 plus a ?100 commission for each television she sells. Company B offers her a position with a salary of ?29,000 plus a ?20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834085038\">\n<div data-type=\"problem\" id=\"fs-id1167834085040\">\n<p id=\"fs-id1167835509988\">Mitchell currently sells stoves for company A at a salary of ?12,000 plus a ?150 commission for each stove he sells. Company B offers him a position with a salary of ?24,000 plus a ?50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835527846\">\n<p id=\"fs-id1167835253891\">Mitchell would need to sell 120 stoves for the companies to be equal.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920428\">\n<div data-type=\"problem\" id=\"fs-id1167831920430\">\n<p id=\"fs-id1167832153327\">Two containers of gasoline hold a total of fifty gallons. The big container can hold ten gallons less than twice the small container. How many gallons does each container hold?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826788597\">\n<div data-type=\"problem\" id=\"fs-id1167834246873\">\n<p id=\"fs-id1167826828264\">June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826880014\">\n<p id=\"fs-id1167826880016\">8 and 40 gallons<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395746\">\n<div data-type=\"problem\" id=\"fs-id1167835479730\">\n<p id=\"fs-id1167835479732\">Shelly spent 10 minutes jogging and 20 minutes cycling and burned 300 calories. The next day, Shelly swapped times, doing 20 minutes of jogging and 10 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834130681\">\n<div data-type=\"problem\" id=\"fs-id1167835361143\">\n<p id=\"fs-id1167835361146\">Drew burned 1800 calories Friday playing one hour of basketball and canoeing for two hours. Saturday he spent two hours playing basketball and three hours canoeing and burned 3200 calories. How many calories did he burn per hour when playing basketball? How many calories did he burn per hour when canoeing?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826987517\">\n<p id=\"fs-id1167826987519\">1000 calories playing basketball and 400 calories canoeing<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835353325\">\n<div data-type=\"problem\" id=\"fs-id1167831036987\">\n<p id=\"fs-id1167831036989\">Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for ?116. Lisa bought two notebooks and three thumb dives for ?68. Find the cost of each notebook and each thumb drive.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832053097\">\n<div data-type=\"problem\" id=\"fs-id1167832066087\">\n<p id=\"fs-id1167832066089\">Nancy bought seven pounds of oranges and three pounds of bananas for ?17. Her husband later bought three pounds of oranges and six pounds of bananas for ?12. What was the cost per pound of the oranges and the bananas?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832053548\">\n<p id=\"fs-id1167830898572\">Oranges cost ?2 per pound and bananas cost ?1 per pound<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535330\">\n<div data-type=\"problem\" id=\"fs-id1167834535332\">\n<p id=\"fs-id1167834367247\">Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for ?114 or she is able to buy 2 shirts and 4 sweaters for ?164. How much does a shirt cost? How much does a sweater cost?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834422585\">\n<div data-type=\"problem\" id=\"fs-id1167832153537\">\n<p id=\"fs-id1167834539431\">Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for ?40 or he is able to buy 5 packages of paper and 6 staplers for ?62. How much does a package of paper cost? How much does a stapler cost?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827943703\">\n<p id=\"fs-id1167827943705\">Package of paper ?4, stapler ?7<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835374673\">\n<div data-type=\"problem\" id=\"fs-id1167831922575\">\n<p id=\"fs-id1167831922577\">The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830699166\">\n<div data-type=\"problem\" id=\"fs-id1167830699168\">\n<p id=\"fs-id1167826940670\">The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835234228\">\n<p id=\"fs-id1167835354843\">Hot dog 150 calories, cup of cottage cheese 220 calories<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834403211\">\n<div data-type=\"problem\" id=\"fs-id1167835309435\">\n<p id=\"fs-id1167835417541\">Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water as juice. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880334\">\n<div data-type=\"problem\" id=\"fs-id1167834191245\">\n<p id=\"fs-id1167834438919\">Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835360066\">\n<p id=\"fs-id1167835360068\">Owen will need 80 quarts of water and 20 quarts of concentrate to make 100 quarts of lemonade.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831871616\"><strong data-effect=\"bold\">Solve Geometry Applications<\/strong><\/p>\n<p id=\"fs-id1167835533545\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832042398\">\n<div data-type=\"problem\" id=\"fs-id1167832042400\">\n<p>The difference of two complementary angles is 55 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835530147\">\n<div data-type=\"problem\" id=\"fs-id1167835530149\">\n<p id=\"fs-id1167834192549\">The difference of two complementary angles is 17 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834079388\">\n<p id=\"fs-id1167831811616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5df404ce5bc65fa13a13b0c10aa45e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> degrees and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d4b3af9422fbd469cebb4e9bf1299df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> degrees<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831908441\">\n<div data-type=\"problem\" id=\"fs-id1167831908443\">\n<p id=\"fs-id1167832066177\">Two angles are complementary. The measure of the larger angle is twelve less than twice the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512309\">\n<div data-type=\"problem\" id=\"fs-id1167834396369\">\n<p id=\"fs-id1167834396372\">Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281915\">\n<p id=\"fs-id1167835229140\">16 degrees and 74 degrees<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057169\">\n<div data-type=\"problem\" id=\"fs-id1167830705518\">\n<p id=\"fs-id1167826987407\">The difference of two supplementary angles is 8 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301936\">\n<div data-type=\"problem\" id=\"fs-id1167835345294\">\n<p id=\"fs-id1167834423315\">The difference of two supplementary angles is 88 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832153564\">\n<p id=\"fs-id1167834279372\">134 degrees and 46 degrees<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831955188\">\n<div data-type=\"problem\" id=\"fs-id1167832128577\">\n<p id=\"fs-id1167834309613\">Two angles are supplementary. The measure of the larger angle is four more than three times the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536390\">\n<div data-type=\"problem\" id=\"fs-id1167834536392\">\n<p id=\"fs-id1167827957061\">Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828420936\">\n<p>37 degrees and 143 degrees<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835373672\">\n<div data-type=\"problem\" id=\"fs-id1167835349809\">\n<p id=\"fs-id1167835349811\">The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826779084\">\n<div data-type=\"problem\" id=\"fs-id1167831970020\">\n<p id=\"fs-id1167831970023\">The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219718\">\n<p id=\"fs-id1167835512843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8f9f4e79c49c9b9f1d869a56ad41292_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a8ae94ba058365d73fa2323538065d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834193531\">\n<div data-type=\"problem\" id=\"fs-id1167834193534\">\n<p id=\"fs-id1167835622049\">The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799204\">\n<div data-type=\"problem\" id=\"fs-id1167834555360\">\n<p id=\"fs-id1167830960829\">The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834524965\">\n<p id=\"fs-id1167835263570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e104257ca81ccb9329a0b9fa023fdba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e104257ca81ccb9329a0b9fa023fdba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835489374\">\n<div data-type=\"problem\" id=\"fs-id1167828411026\">\n<p id=\"fs-id1167828411028\">Wayne is hanging a string of lights 45 feet long around the three sides of his patio, which is adjacent to his house. The length of his patio, the side along the house, is five feet longer than twice its width. Find the length and width of the patio.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835335808\">\n<div data-type=\"problem\" id=\"fs-id1167835335810\">\n<p id=\"fs-id1167835241096\">Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his front yard. The length is five feet less than three times the width. Find the length and width of the fencing.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834377414\">\n<p id=\"fs-id1167834377416\">Width is 41 feet and length is 118 feet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865390\">\n<div data-type=\"problem\" id=\"fs-id1167831959127\">\n<p id=\"fs-id1167831959129\">A frame around a family portrait has a perimeter of 90 inches. The length is fifteen less than twice the width. Find the length and width of the frame.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194513\">\n<div data-type=\"problem\" id=\"fs-id1167835194860\">\n<p id=\"fs-id1167835375429\">The perimeter of a toddler play area is 100 feet. The length is ten more than three times the width. Find the length and width of the play area.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835190449\">\n<p id=\"fs-id1167831888088\">Width is 10 feet and length is 40 feet.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830697840\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\n<p id=\"fs-id1167835353585\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835334297\">\n<div data-type=\"problem\" id=\"fs-id1167835479058\">\n<p id=\"fs-id1167835173769\">Sarah left Minneapolis heading east on the interstate at a speed of 60 mph. Her sister followed her on the same route, leaving two hours later and driving at a rate of 70 mph. How long will it take for Sarah\u2019s sister to catch up to Sarah?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834330007\">\n<div data-type=\"problem\" id=\"fs-id1167835510716\">\n<p id=\"fs-id1167835198385\">College roommates John and David were driving home to the same town for the holidays. John drove 55 mph, and David, who left an hour later, drove 60 mph. How long will it take for David to catch up to John?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831985703\">\n<p id=\"fs-id1167834539187\">11 hours<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978939\">\n<div data-type=\"problem\" id=\"fs-id1167826978942\">\n<p id=\"fs-id1167834120674\">At the end of spring break, Lucy left the beach and drove back towards home, driving at a rate of 40 mph. Lucy\u2019s friend left the beach for home 30 minutes (half an hour) later, and drove 50 mph. How long did it take Lucy\u2019s friend to catch up to Lucy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835305691\">\n<div data-type=\"problem\" id=\"fs-id1167835305693\">\n<p id=\"fs-id1167835355663\">Felecia left her home to visit her daughter driving 45 mph. Her husband waited for the dog sitter to arrive and left home twenty minutes (1\/3 hour) later. He drove 55 mph to catch up to Felecia. How long before he reaches her?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834324665\">\n<p id=\"fs-id1167834324667\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b2514947ef7a7471f9fcd9c60117cf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"21\" style=\"vertical-align: -1px;\" \/> hour<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835231291\">\n<div data-type=\"problem\" id=\"fs-id1167835231293\">\n<p id=\"fs-id1167832211867\">The Jones family took a 12-mile canoe ride down the Indian River in two hours. After lunch, the return trip back up the river took three hours. Find the rate of the canoe in still water and the rate of the current.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834593438\">\n<div data-type=\"problem\" id=\"fs-id1167834593441\">\n<p id=\"fs-id1167834532384\">A motor boat travels 60 miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831162731\">Boat rate is 16 mph and current rate is 4 mph.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831081632\">\n<div data-type=\"problem\" id=\"fs-id1167835373743\">\n<p id=\"fs-id1167835373746\">A motor boat traveled 18 miles down a river in two hours but going back upstream, it took <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fac357af10206befdb463e651007c5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"22\" style=\"vertical-align: -1px;\" \/> hours due to the current. Find the rate of the motor boat in still water and the rate of the current. (Round to the nearest hundredth.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057264\">\n<div data-type=\"problem\" id=\"fs-id1167832057266\">\n<p id=\"fs-id1167834533568\">A river cruise boat sailed 80 miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831191465\">\n<p id=\"fs-id1167831837698\">Boat rate is 18 mph and current rate is 2 mph.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831228810\">\n<div data-type=\"problem\" id=\"fs-id1167831228812\">\n<p id=\"fs-id1167835274915\">A small jet can fly 1072 miles in 4 hours with a tailwind but only 848 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120690\">\n<div data-type=\"problem\" id=\"fs-id1167834432334\">\n<p id=\"fs-id1167834432337\">A small jet can fly 1435 miles in 5 hours with a tailwind but only 1,215 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834094759\">\n<p id=\"fs-id1167834094761\">Jet rate is 265 mph and wind speed is 22 mph.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832066764\">\n<div data-type=\"problem\" id=\"fs-id1167835498833\">\n<p id=\"fs-id1167835498835\">A commercial jet can fly 868 miles in 2 hours with a tailwind but only 792 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835312119\">\n<div data-type=\"problem\" id=\"fs-id1167835312121\">\n<p id=\"fs-id1167830865979\">A commercial jet can fly 1,320 miles in 3 hours with a tailwind but only 1170 miles in 3 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306122\">\n<p id=\"fs-id1167835226254\">Jet rate is 415 mph and wind speed is 25 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835321214\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835283060\">\n<div data-type=\"problem\" id=\"fs-id1167835283062\">\n<p id=\"fs-id1167834161535\">Write an application problem similar to <a href=\"#fs-id1167834556567\" class=\"autogenerated-content\">(Figure)<\/a>. Then translate to a system of equations and solve it.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912147\">\n<div data-type=\"problem\" id=\"fs-id1167831912149\">\n<p id=\"fs-id1167831106889\">Write a uniform motion problem similar to <a href=\"#fs-id1167835186838\" class=\"autogenerated-content\">(Figure)<\/a> that relates to where you live with your friends or family members. Then translate to a system of equations and solve it.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834190356\">\n<p id=\"fs-id1167835230726\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835311989\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834489878\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834324745\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve direct translation applications, solve geometry applications, solve uniform motion applications. The remaining columns are empty.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve direct translation applications, solve geometry applications, solve uniform motion applications. The remaining columns are empty.\" \/><\/span><\/p>\n<p id=\"fs-id1167828426742\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167834183572\">\n<dt>complementary angles<\/dt>\n<dd id=\"fs-id1167832010323\">Two angles are complementary if the sum of the measures of their angles is 90 degrees.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167835623490\">\n<dt>supplementary angles<\/dt>\n<dd>Two angles are supplementary if the sum of the measures of their angles is 180 degrees.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2436","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2436\/revisions"}],"predecessor-version":[{"id":15234,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2436\/revisions\/15234"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2305"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}