{"id":116,"date":"2022-04-13T10:02:15","date_gmt":"2022-04-13T10:02:15","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=116"},"modified":"2024-12-20T10:49:08","modified_gmt":"2024-12-20T10:49:08","slug":"e-9-solve-a-pair-of-equations-using-elimination-word-problems","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/e-9-solve-a-pair-of-equations-using-elimination-word-problems\/","title":{"rendered":"E.9 Solve a pair of equations using elimination: word problems"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color has-huge-font-size\" style=\"color:#00056d;text-transform:uppercase\"><strong>Solve a pair of equations using elimination: word problems<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-ac2bc047fac346cd34a46d2a76287634\" style=\"color:#74008b;text-transform:capitalize\">key notes:<\/p>\n\n\n\n<p class=\"has-large-font-size\">To solve using elimination, follow these four steps:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">Step 1: Make sure the equations have opposite&nbsp;<em>x<\/em>&nbsp;terms or opposite&nbsp;<em>y<\/em>&nbsp;terms.<\/li>\n\n\n\n<li class=\"has-large-font-size\">Step 2: Add to eliminate one variable and solve for the other.<\/li>\n\n\n\n<li class=\"has-large-font-size\">Step 3: Plug the result of Step 2 into one of the original equations and solve.<\/li>\n\n\n\n<li class=\"has-large-font-size\">Step 4: State the solution.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#ede191\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.<\/p>\n\n\n\n<p>An administrative assistant is making some copies. She made 23 one-sided copies and 48 two-sided copies for the V.P. of Marketing, which took a total of 286 seconds. Next, she made 13 one-sided copies and 24 two-sided copies for the Director of Sales, which took 146 seconds.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-5b4d65b75cc44c3341b48eecc75f316c\" style=\"color:#b00012\"> How long does it take to make each type of copy?<\/p>\n\n\n\n<p>It takes&nbsp;&nbsp;_____seconds to make a one-sided copy and&nbsp;&nbsp;_____seconds to make a two-sided copy.<\/p>\n<\/div><\/div>\n\n\n\n<p>Before you can solve, you must write simultaneous equations. Let&nbsp;<em>x<\/em>&nbsp;represent the amount of time to make a one-sided copy, and let&nbsp;<em>y<\/em>&nbsp;represent the amount of time to make a two-sided copy.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>23<em>x<\/em>&nbsp;+ 48<em>y<\/em>&nbsp;= 286<\/li>\n\n\n\n<li>13<em>x<\/em>&nbsp;+ 24<em>y<\/em>&nbsp;= 146<\/li>\n<\/ul>\n\n\n\n<p>Now use elimination to solve the simultaneous equations.<\/p>\n\n\n\n<p><strong>Step 1: Make sure the equations have opposite&nbsp;<em>x<\/em>&nbsp;terms or opposite&nbsp;<em>y<\/em>&nbsp;terms.<\/strong><\/p>\n\n\n\n<p>Currently, neither the&nbsp;<em>x<\/em>&nbsp;terms (23<em>x<\/em>&nbsp;and 13<em>x<\/em>) nor the&nbsp;<em>y<\/em>&nbsp;terms (48<em>y<\/em>&nbsp;and 24<em>y<\/em>) are opposites. Use multiplication to rewrite the equations with either opposite&nbsp;<em>x<\/em>&nbsp;terms or opposite&nbsp;<em>y<\/em>&nbsp;terms. One good approach is to multiply the second equation by&nbsp;-2.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>23<em>x<\/em>&nbsp;+ 48<em>y<\/em>&nbsp;= 286<\/td><td>&nbsp;\u2192&nbsp;<\/td><td>23<em>x<\/em>&nbsp;+ 48<em>y<\/em>&nbsp;= 286<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure id=\"yui_3_18_1_1_1670055548835_350\" class=\"wp-block-table\"><table><tbody><tr><td><strong>-2<\/strong>(13<em>x<\/em>&nbsp;+ 24<em>y<\/em>&nbsp;= 146)<\/td><td>&nbsp;\u2192&nbsp;<\/td><td><strong>-26<\/strong><em>x<\/em>&nbsp;\u2212&nbsp;<strong>48<\/strong><em>y<\/em>&nbsp;=&nbsp;<strong>-292<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Now the&nbsp;<em>y<\/em>&nbsp;terms (48<em>y<\/em>&nbsp;and&nbsp;-48<em>y<\/em>) are opposites.<\/p>\n\n\n\n<p><strong>Step 2: Add to eliminate one variable and solve for the other.<\/strong><\/p>\n\n\n\n<p>Add to eliminate the&nbsp;<em>y<\/em>&nbsp;terms, and then solve for&nbsp;<em>x<\/em>.<\/p>\n\n\n\n<p style=\"line-height:0\">       23x + 48y = 286<\/p>\n\n\n\n<p style=\"line-height:0\">+ <span style=\"text-decoration: underline;\">( \u201326x \u2212 48y = \u2013292 )<\/span><\/p>\n\n\n\n<p style=\"line-height:0\"><br>         \u20133x + 0y = \u20136         &#8212;&#8212;-&gt; Add to eliminate the y terms<\/p>\n\n\n\n<p>                \u20133x = \u20136 Simplify<\/p>\n\n\n\n<p>                x = 2 Divide both sides by \u20133<\/p>\n\n\n\n<p><strong>Step 3: Plug the result of Step 2 into one of the original equations and solve.<\/strong><\/p>\n\n\n\n<p>Take the result of Step 2,&nbsp;<em>x<\/em>&nbsp;= 2, and plug it into one of the original equations, such as&nbsp;23<em>x<\/em>&nbsp;+ 48<em>y<\/em>&nbsp;= 286. Then find the value of&nbsp;<em>y<\/em>.<\/p>\n\n\n\n<p>23<strong><em>x<\/em><\/strong>&nbsp;+ 48<em>y<\/em>&nbsp;=&nbsp;286<br>23(<strong>2<\/strong>)&nbsp;+ 48y&nbsp;=&nbsp;286    &#8212;&gt;Plug in&nbsp;x&nbsp;= 2<br>46 + 48y&nbsp;=&nbsp;286       &#8212;-&gt;Multiply<br>48y&nbsp;=&nbsp;240       &#8212;-&gt;Subtract 46 from both sides<br>y&nbsp;=&nbsp;5   &#8212;&#8211;&gt;Divide both sides by 48<\/p>\n\n\n\n<p><strong>Step 4: State the solution.<\/strong><\/p>\n\n\n\n<p>Since&nbsp;<em>x<\/em>&nbsp;= 2 and&nbsp;<em>y<\/em>&nbsp;= 5, the solution is&nbsp;(2,&nbsp;5).<\/p>\n\n\n\n<p>It takes 2 seconds to make a one-sided copy and 5 seconds to make a two-sided copy.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#9ff3e7\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.<\/p>\n\n\n\n<p>Shannon is selling her handmade jewellery online. Yesterday, she sold 10 bracelets and 4 necklaces, for a profit of&nbsp;\u20b9188. Today, she made a profit of&nbsp;\u20b974 by selling 3 bracelets and 2 necklaces. <\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-6cc37233de9436e22335923a9dd420d8\" style=\"color:#b00012\">How much profit does Shannon earn from each piece?<\/p>\n\n\n\n<p>Shannon earns a profit of&nbsp;\u20b9_____&nbsp;from every bracelet and&nbsp;\u20b9____&nbsp;from every necklace.<\/p>\n<\/div><\/div>\n\n\n\n<p>Before you can solve, you must write simultaneous equations. Let&nbsp;<em>x<\/em>&nbsp;represent the profit on every bracelet, and let&nbsp;<em>y<\/em>&nbsp;represent the profit on every necklace.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>10<em>x<\/em>&nbsp;+ 4<em>y<\/em>&nbsp;= 188<\/li>\n\n\n\n<li>3<em>x<\/em>&nbsp;+ 2<em>y<\/em>&nbsp;= 74<\/li>\n<\/ul>\n\n\n\n<p>Now use elimination to solve the simultaneous equations.<\/p>\n\n\n\n<p><strong>Step 1: Make sure the equations have opposite&nbsp;<em>x<\/em>&nbsp;terms or opposite&nbsp;<em>y<\/em>&nbsp;terms.<\/strong><\/p>\n\n\n\n<p>Currently, neither the&nbsp;<em>x<\/em>&nbsp;terms (10<em>x<\/em>&nbsp;and 3<em>x<\/em>) nor the&nbsp;<em>y<\/em>&nbsp;terms (4<em>y<\/em>&nbsp;and 2<em>y<\/em>) are opposites. Use multiplication to rewrite the equations with either opposite&nbsp;<em>x<\/em>&nbsp;terms or opposite&nbsp;<em>y<\/em>&nbsp;terms. One good approach is to multiply the second equation by&nbsp;<sup>\u2013<\/sup>2.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>10<em>x<\/em>&nbsp;+ 4<em>y<\/em>&nbsp;= 188<\/td><td>&nbsp;\u2192&nbsp;<\/td><td>10<em>x<\/em>&nbsp;+ 4<em>y<\/em>&nbsp;= 188<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>-2<\/strong>(3<em>x<\/em>&nbsp;+ 2<em>y<\/em>&nbsp;= 74)<\/td><td>&nbsp;\u2192&nbsp;<\/td><td><strong>-6<\/strong><em>x<\/em>&nbsp;\u2212&nbsp;<strong>4<\/strong><em>y<\/em>&nbsp;=&nbsp;<strong>-148<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Now the&nbsp;<em>y<\/em>&nbsp;terms (4<em>y<\/em>&nbsp;and&nbsp;-4<em>y<\/em>) are opposites.<\/p>\n\n\n\n<p><strong>Step 2: Add to eliminate one variable and solve for the other.<\/strong><\/p>\n\n\n\n<p>Add to eliminate the&nbsp;<em>y<\/em>&nbsp;terms, and then solve for&nbsp;<em>x<\/em>.<\/p>\n\n\n\n<p style=\"line-height:0\">         10x + 4y = 188<\/p>\n\n\n\n<p style=\"line-height:0\">+    <span style=\"text-decoration: underline;\">( \u20136x \u2212 4y = \u2013148 )<\/span><\/p>\n\n\n\n<p style=\"line-height:0\"><br>          4x + 0y = 40     &#8212;&gt; Add to eliminate the y terms<\/p>\n\n\n\n<p>          4x = 40     &#8212;&#8212;-&gt;Simplify<\/p>\n\n\n\n<p>            x = 10       &#8212;&#8212;-&gt; Divide both sides by 4<\/p>\n\n\n\n<p><strong>Step 3: Plug the result of Step 2 into one of the original equations and solve.<\/strong><\/p>\n\n\n\n<p>Take the result of Step 2,&nbsp;<em>x<\/em>&nbsp;= 10, and plug it into one of the original equations, such as&nbsp;10<em>x<\/em>&nbsp;+ 4<em>y<\/em>&nbsp;= 188. Then find the value of&nbsp;<em>y<\/em>.<\/p>\n\n\n\n<p style=\"line-height:1.9\">10<strong><em>x<\/em><\/strong>&nbsp;+ 4<em>y<\/em>&nbsp;=&nbsp;188<\/p>\n\n\n\n<p style=\"line-height:1.5\">10(<strong>10<\/strong>)&nbsp;+ 4y&nbsp;=&nbsp;188     &#8212;&gt; Plug in&nbsp;x&nbsp;= 10<br>100 + 4y&nbsp;=&nbsp;188      &#8212;-&gt; Multiply<br>4y&nbsp;=&nbsp;88     &#8212;-&gt; Subtract 100 from both sides<br>y&nbsp;=&nbsp;22      &#8212;-&gt; Divide both sides by 4<\/p>\n\n\n\n<p><strong>Step 4: State the solution.<\/strong><\/p>\n\n\n\n<p>Since&nbsp;<em>x<\/em>&nbsp;= 10 and&nbsp;<em>y<\/em>&nbsp;= 22, the solution is&nbsp;(10,&nbsp;22).<\/p>\n\n\n\n<p>Shannon earns a profit of&nbsp;\u20b910&nbsp;from every bracelet and&nbsp;\u20b922&nbsp;from every necklace.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f6b8d4\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks.<\/p>\n\n\n\n<p>Tina and Elise decided to shoot arrows at a simple target with a large outer ring and a smaller bull&#8217;s-eye. Tina went first and landed 4 arrows in the outer ring and 3 arrows in the bull&#8217;s-eye, for a total of 307 points. Elise went second and got 1 arrow in the outer ring and 3 arrows in the bull&#8217;s-eye, earning a total of 241 points.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-2890f3e8f8f9763c55d97934e70cab56\" style=\"color:#b00012\"> How many points is each region of the target worth?<\/p>\n\n\n\n<p>The outer ring is worth&nbsp;_______&nbsp;points, and the bull&#8217;s-eye is worth&nbsp;___&nbsp;points.<\/p>\n<\/div><\/div>\n\n\n\n<p>Before you can solve, you must write simultaneous equations. Let x represent the points awarded for an arrow in the outer ring, and let y represent the points awarded for an arrow in the bulls-eye.<\/p>\n\n\n\n<p>4x + 3y = 307<br>x + 3y = 241<\/p>\n\n\n\n<p>Now use elimination to solve the simultaneous equations.<\/p>\n\n\n\n<p>Step 1: Make sure the equations have opposite x terms or opposite y terms.<\/p>\n\n\n\n<p>Currently, neither the x terms (4x and x) nor the y terms (3y and 3y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the first equation by \u20131.<\/p>\n\n\n\n<p>\u2013(4x + 3y = 307)\u2192 \u20134x \u2212 3y = \u2013307<br>x + 3y = 241\u2192 x + 3y = 241<\/p>\n\n\n\n<p>Now the y terms (\u20133y and 3y) are opposites.<\/p>\n\n\n\n<p>Step 2: Add to eliminate one variable and solve for the other.<\/p>\n\n\n\n<p>Add to eliminate the y terms, and then solve for x.<\/p>\n\n\n\n<p>\u20134x \u2212 3y = \u2013307<\/p>\n\n\n\n<p><span style=\"text-decoration: underline;\">+( x + 3y = 241 )<\/span><br>\u20133x + 0y = \u201366    Add to eliminate the y terms<\/p>\n\n\n\n<p>\u20133x = \u201366 Simplify<\/p>\n\n\n\n<p>x = 22 Divide both sides by \u20133<\/p>\n\n\n\n<p>Step 3: Plug the result of Step 2 into one of the original equations and solve.<\/p>\n\n\n\n<p>Take the result of Step 2, x = 22, and plug it into one of the original equations, such as 4x + 3y = 307. Then find the value of y.<\/p>\n\n\n\n<p>4x + 3y = 307<\/p>\n\n\n\n<p>4(22) + 3y = 307 Plug in x = 22<\/p>\n\n\n\n<p>88 + 3y = 307 Multiply<\/p>\n\n\n\n<p>3y = 219 Subtract 88 from both sides<\/p>\n\n\n\n<p>y = 73 Divide both sides by 3<\/p>\n\n\n\n<p>Step 4: State the solution.<\/p>\n\n\n\n<p>Since x = 22 and y = 73, the solution is (22, 73).<\/p>\n\n\n\n<p>The outer ring is worth 22 points, and the bull&#8217;s-eye is worth 73 points.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">let&#8217;s practice:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full is-resized\"><a href=\"https:\/\/wordwall.net\/play\/38993\/783\/420\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-32.png\" alt=\"\" class=\"wp-image-6654\" style=\"width:268px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-32.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-32-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-32-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure><\/div>\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/wordwall.net\/play\/84110\/826\/505\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-7.png\" alt=\"\" class=\"wp-image-10989\" style=\"width:289px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-7.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-7-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-7-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Solve a pair of equations using elimination: word problems key notes: To solve using elimination, follow these four steps: Learn with an example Write simultaneous equations to describe the situation below, solve using elimination, and fill in the blanks. An administrative assistant is making some copies. She made 23 one-sided copies and 48 two-sided copies<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/e-9-solve-a-pair-of-equations-using-elimination-word-problems\/\">Continue reading <span class=\"screen-reader-text\">&#8220;E.9 Solve a pair of equations using elimination: word problems&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-116","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":32,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116\/revisions"}],"predecessor-version":[{"id":15447,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/116\/revisions\/15447"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}