{"id":3179,"date":"2025-08-15T22:29:18","date_gmt":"2025-08-15T22:29:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3179"},"modified":"2026-03-20T16:26:06","modified_gmt":"2026-03-20T16:26:06","slug":"systems-of-linear-equations-three-variables-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-three-variables-apply-it\/","title":{"raw":"Systems of Linear Equations: Three Variables: Apply It","rendered":"Systems of Linear Equations: Three Variables: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve systems of three equations in three variables.<\/li>\r\n \t<li>Identify inconsistent systems of equations containing three variables.<\/li>\r\n \t<li>Express the solution of a system of dependent equations containing three variables.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><\/div>\r\n<\/div>\r\n<div class=\"h-8\">\r\n<div class=\"h-8\">\r\n<p class=\"whitespace-normal break-words\">Maya just graduated from college and landed her first job with a starting bonus of $8,400. She wants to invest this money wisely across three different investment options to diversify her portfolio and maximize her returns while managing risk.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Investing is a process of using your money to buy something (stocks, real estate, bitcoin, etc) that you expect will be worth more in the future.Thanks to compound interest, when your money earns money, and then that new money earns money too, your investments can grow quickly. A $1,000 investment at age 20 could become $20,000+ by retirement, but the same investment at age 40 might only become $7,000.Learn more about investing at <a href=\"https:\/\/www.dowjanes.com\/blog\/blog-investing-in-the-stock-market-beginners-start-investing?gc_id=22694557882&amp;h_ga_id=181062063573&amp;h_ad_id=759240638084&amp;h_keyword_id=kwd-2319357952547&amp;h_keyword=online%20course%20on%20financial%20literacy&amp;h_placement=&amp;gad_source=1&amp;gad_campaignid=22694557882&amp;gbraid=0AAAAACRF2f-jfntUR_xGHMxNQmjaMd9Vv&amp;gclid=Cj0KCQjw267GBhCSARIsAOjVJ4EuWl9ggnacECIGjCjJ4DUzZcDO63Vff5XkMNcswL5EK4TyM_xzhskaAj-pEALw_wcB\">dowjanes.com.<\/a><\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Investment Options<\/h2>\r\n<p class=\"whitespace-normal break-words\">After researching different investment vehicles, Maya has identified three options:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">High Yield Savings Account: 3% annual interest (low risk)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Money Market Account: 6% annual return (medium risk)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stock Market Index Funds: 9% annual return (higher risk)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Maya has set some guidelines for her investment strategy:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">She wants to invest twice as much in the stock market index funds as in the high yield savings account to take advantage of higher returns while she's young<\/li>\r\n \t<li class=\"whitespace-normal break-words\">She wants to earn $540 in total returns during the first year<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Let:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex] = amount invested in high yield savings account (3%)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex] = amount invested in money market account (6%)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z[\/latex] = amount invested in stock market index funds (9%)<\/li>\r\n<\/ul>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The interest earned in one year can be calculated using simple interest: [latex]I=Prt[\/latex]. Where [latex]P[\/latex] is the initial investment, [latex]r[\/latex] is the interest rate as a decimal, and [latex]t[\/latex] is fixed at [latex]1[\/latex] year.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312315[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the investment system.[reveal-answer q=\"180715\"]Show step 1[\/reveal-answer]\r\n[hidden-answer a=\"180715\"][latex]\\begin{align} x + y + z &amp;= 8400 &amp;&amp; (1)\\\\ z &amp;= 2y &amp;&amp; (2)\\\\ 0.03x + 0.06y + 0.09z &amp;= 540 &amp;&amp; (3) \\end{align}[\/latex]Use substitution with Equation (2) and substitute [latex]2y[\/latex] for [latex]z[\/latex] in Equation (1) and Equation (3):\r\n<ul>\r\n \t<li>Equation 1:[latex]\\begin{align} x + 2z + z &amp;= 8400 &amp;&amp;\\text{ Substitute} \\\\ x+3z &amp;= 8400 &amp;&amp;\\text{ Combine } 2z+z\\end{align}[\/latex]<\/li>\r\n \t<li>Equation 3:[latex]\\begin{align} 0.03x + 0.06(2z) + 0.09z &amp;= 540 &amp;&amp;\\text{ Substitute } \\\\ 0.03x+0.12z +0.09z &amp;= 540 &amp;&amp; \\text{ Multiply } 0.06(2z) \\\\ 0.03x + 0.21z &amp;= 540 &amp;&amp; \\text{ Combine } 0.12z + 0.09z \\end{align}[\/latex]<\/li>\r\n<\/ul>\r\nWe now have a new [latex]2\\times 2[\/latex] system to solve:\r\n\r\n[latex]\\begin{align} x + 3z &amp;= 8400 &amp;&amp; (4)\\\\ 0.03x + 0.21z\u00a0 &amp;= 540 &amp;&amp; (5) \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"392311\"]Show step 2[\/reveal-answer]\r\n[hidden-answer a=\"392311\"]\r\n\r\nUse elimination method to solve the new system:\r\n\r\n[latex]\\begin{align} x + 3z &amp;= 8400 &amp;&amp;(4) \\\\ -3x -21z\u00a0 &amp;= -54000&amp;&amp; \\text{ Multiply (5) by } -100\\end{align}[\/latex]\r\n\r\n[latex]\\begin{align} 3x + 9z &amp;= 25200 &amp;&amp; \\text{ Multiply (4) by }3 \\\\ -3x -21z\u00a0 &amp;= -54000\\end{align}[\/latex]\r\n\r\n[latex]\\begin{align} 3x + 9z &amp;= 25200 \\\\ -3x -21z\u00a0 &amp;= -54000 \\\\ -12z &amp;= -28,800 &amp;&amp; \\text{ Add the equations to eliminate } x \\end{align}[\/latex]\r\n\r\nSolve the equation: [latex]-12x=-28,800[\/latex]\r\n\r\n[latex]\\begin{align} -12x&amp;=-28800 \\\\ x &amp;= 2400 &amp;&amp; \\text{ Divide by } -12 \\end{align}[\/latex]\r\n\r\nSubstitute [latex]x=2400[\/latex] back into Equation (4):\r\n\r\n[latex]\\begin{align} x+3z&amp;=8400 \\\\2400 + 3z\u00a0 &amp;= 8400 &amp;&amp; \\text{Substitute} \\\\ 3z &amp;= 6000 &amp;&amp; \\text{Subtract } 2400 \\\\ z&amp;=2000 &amp;&amp; \\text{Divide by } 3 \\end{align}[\/latex]\r\n\r\nFinally, substitute [latex]x[\/latex] and [latex]z[\/latex] into Equation (1).\r\n\r\n\\begin{align} x + y + z &amp;= 8400 \\\\ 2400 + y + 2000 &amp;= 8400 &amp;&amp; \\text{ Substitute } \\\\ y + 4400 &amp;= 8400 &amp;&amp; \\text{ Simplify } \\\\ y &amp;= 4000 &amp;&amp; \\text{ Subtract } 4400 \\end{align}[\/latex]\r\n\r\n[hidden-answer a=\"392311\"]\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"423798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"423798\"]Maya should invest $2,400 in her high yield savings account, $2,000 in her stock market portfolio, and $4,000 in her money market account.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox youChoose\" aria-label=\"You Choose\">[choosedataset divId=\"tnh-choose-dataset\" title=\"Choose Your Own Investment Plan\" label=\"\" default=\"Choose a Starting Investment\"][datasetoption]\r\n[displayname]Investing $1,100[\/displayname]\r\n[ohmid]42420[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Investing $10,000[\/displayname]\r\n[ohmid]42421[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Investing $50,000[\/displayname]\r\n[ohmid]42422[\/ohmid]\r\n[\/datasetoption]\r\n[\/choosedataset]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve systems of three equations in three variables.<\/li>\n<li>Identify inconsistent systems of equations containing three variables.<\/li>\n<li>Express the solution of a system of dependent equations containing three variables.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><\/div>\n<\/div>\n<div class=\"h-8\">\n<div class=\"h-8\">\n<p class=\"whitespace-normal break-words\">Maya just graduated from college and landed her first job with a starting bonus of $8,400. She wants to invest this money wisely across three different investment options to diversify her portfolio and maximize her returns while managing risk.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Investing is a process of using your money to buy something (stocks, real estate, bitcoin, etc) that you expect will be worth more in the future.Thanks to compound interest, when your money earns money, and then that new money earns money too, your investments can grow quickly. A $1,000 investment at age 20 could become $20,000+ by retirement, but the same investment at age 40 might only become $7,000.Learn more about investing at <a href=\"https:\/\/www.dowjanes.com\/blog\/blog-investing-in-the-stock-market-beginners-start-investing?gc_id=22694557882&amp;h_ga_id=181062063573&amp;h_ad_id=759240638084&amp;h_keyword_id=kwd-2319357952547&amp;h_keyword=online%20course%20on%20financial%20literacy&amp;h_placement=&amp;gad_source=1&amp;gad_campaignid=22694557882&amp;gbraid=0AAAAACRF2f-jfntUR_xGHMxNQmjaMd9Vv&amp;gclid=Cj0KCQjw267GBhCSARIsAOjVJ4EuWl9ggnacECIGjCjJ4DUzZcDO63Vff5XkMNcswL5EK4TyM_xzhskaAj-pEALw_wcB\">dowjanes.com.<\/a><\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Investment Options<\/h2>\n<p class=\"whitespace-normal break-words\">After researching different investment vehicles, Maya has identified three options:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">High Yield Savings Account: 3% annual interest (low risk)<\/li>\n<li class=\"whitespace-normal break-words\">Money Market Account: 6% annual return (medium risk)<\/li>\n<li class=\"whitespace-normal break-words\">Stock Market Index Funds: 9% annual return (higher risk)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Maya has set some guidelines for her investment strategy:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">She wants to invest twice as much in the stock market index funds as in the high yield savings account to take advantage of higher returns while she&#8217;s young<\/li>\n<li class=\"whitespace-normal break-words\">She wants to earn $540 in total returns during the first year<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Let:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex] = amount invested in high yield savings account (3%)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex] = amount invested in money market account (6%)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z[\/latex] = amount invested in stock market index funds (9%)<\/li>\n<\/ul>\n<section class=\"textbox recall\" aria-label=\"Recall\">The interest earned in one year can be calculated using simple interest: [latex]I=Prt[\/latex]. Where [latex]P[\/latex] is the initial investment, [latex]r[\/latex] is the interest rate as a decimal, and [latex]t[\/latex] is fixed at [latex]1[\/latex] year.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312315\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312315&theme=lumen&iframe_resize_id=ohm312315&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the investment system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q180715\">Show step 1<\/button><\/p>\n<div id=\"q180715\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align} x + y + z &= 8400 && (1)\\\\ z &= 2y && (2)\\\\ 0.03x + 0.06y + 0.09z &= 540 && (3) \\end{align}[\/latex]Use substitution with Equation (2) and substitute [latex]2y[\/latex] for [latex]z[\/latex] in Equation (1) and Equation (3):<\/p>\n<ul>\n<li>Equation 1:[latex]\\begin{align} x + 2z + z &= 8400 &&\\text{ Substitute} \\\\ x+3z &= 8400 &&\\text{ Combine } 2z+z\\end{align}[\/latex]<\/li>\n<li>Equation 3:[latex]\\begin{align} 0.03x + 0.06(2z) + 0.09z &= 540 &&\\text{ Substitute } \\\\ 0.03x+0.12z +0.09z &= 540 && \\text{ Multiply } 0.06(2z) \\\\ 0.03x + 0.21z &= 540 && \\text{ Combine } 0.12z + 0.09z \\end{align}[\/latex]<\/li>\n<\/ul>\n<p>We now have a new [latex]2\\times 2[\/latex] system to solve:<\/p>\n<p>[latex]\\begin{align} x + 3z &= 8400 && (4)\\\\ 0.03x + 0.21z\u00a0 &= 540 && (5) \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q392311\">Show step 2<\/button><\/p>\n<div id=\"q392311\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use elimination method to solve the new system:<\/p>\n<p>[latex]\\begin{align} x + 3z &= 8400 &&(4) \\\\ -3x -21z\u00a0 &= -54000&& \\text{ Multiply (5) by } -100\\end{align}[\/latex]<\/p>\n<p>[latex]\\begin{align} 3x + 9z &= 25200 && \\text{ Multiply (4) by }3 \\\\ -3x -21z\u00a0 &= -54000\\end{align}[\/latex]<\/p>\n<p>[latex]\\begin{align} 3x + 9z &= 25200 \\\\ -3x -21z\u00a0 &= -54000 \\\\ -12z &= -28,800 && \\text{ Add the equations to eliminate } x \\end{align}[\/latex]<\/p>\n<p>Solve the equation: [latex]-12x=-28,800[\/latex]<\/p>\n<p>[latex]\\begin{align} -12x&=-28800 \\\\ x &= 2400 && \\text{ Divide by } -12 \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]x=2400[\/latex] back into Equation (4):<\/p>\n<p>[latex]\\begin{align} x+3z&=8400 \\\\2400 + 3z\u00a0 &= 8400 && \\text{Substitute} \\\\ 3z &= 6000 && \\text{Subtract } 2400 \\\\ z&=2000 && \\text{Divide by } 3 \\end{align}[\/latex]<\/p>\n<p>Finally, substitute [latex]x[\/latex] and [latex]z[\/latex] into Equation (1).<\/p>\n<p>\\begin{align} x + y + z &amp;= 8400 \\\\ 2400 + y + 2000 &amp;= 8400 &amp;&amp; \\text{ Substitute } \\\\ y + 4400 &amp;= 8400 &amp;&amp; \\text{ Simplify } \\\\ y &amp;= 4000 &amp;&amp; \\text{ Subtract } 4400 \\end{align}[\/latex]<\/p>\n<div id=\"q392311\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q423798\">Show Solution<\/button><\/p>\n<div id=\"q423798\" class=\"hidden-answer\" style=\"display: none\">Maya should invest $2,400 in her high yield savings account, $2,000 in her stock market portfolio, and $4,000 in her money market account.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<div id=\"tnh-choose-dataset\" class=\"chooseDataset\">\n<h3>Choose Your Own Investment Plan<\/h3>\n<form><select name=\"dataset\"><option value=\"\">Choose a Starting Investment<\/option><option value=\"42420\">Investing $1,100<\/option><option value=\"42421\">Investing $10,000<\/option><option value=\"42422\">Investing $50,000<\/option><\/select><\/form>\n<div class=\"ohmContainer\"><\/div>\n<\/p><\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":[{"divId":"tnh-choose-dataset","title":"Choose Your Own Investment Plan","label":"","default":"Choose a Starting Investment","try_it_collection":[{"displayName":"Investing $1,100","value":"42420"},{"displayName":"Investing $10,000","value":"42421"},{"displayName":"Investing $50,000","value":"42422"}]}],"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3179"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3179\/revisions"}],"predecessor-version":[{"id":5944,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3179\/revisions\/5944"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3179\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3179"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3179"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3179"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}