{"id":4347,"date":"2013-06-29T04:09:24","date_gmt":"2013-06-29T04:09:24","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=4347"},"modified":"2026-03-26T09:50:32","modified_gmt":"2026-03-26T14:50:32","slug":"substitution-and-elimination-for-solving-linear-systems","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/substitution-and-elimination-for-solving-linear-systems\/","title":{"rendered":"Substitution and Elimination for Solving Linear Systems"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_8TNtQHtDmJk\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_8TNtQHtDmJk\" data-source-videoID=\"8TNtQHtDmJk\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Substitution and Elimination for Solving Linear Systems Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Substitution and Elimination for Solving Linear Systems\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_8TNtQHtDmJk:hover {cursor:pointer;} img#videoThumbnailImage_8TNtQHtDmJk {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-solving-systems-of-linear-equations-64c14030155d6-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_8TNtQHtDmJk\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_8TNtQHtDmJk\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Substitution and Elimination for Solving Linear Systems\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_8TNtQHtDmJk\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_8TNtQHtDmJk\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_8TNtQHtDmJk\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Y1R_Function() {\n  var x = document.getElementById(\"Y1R\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"Y1R_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"Y1R\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Substitution_Method\" class=\"smooth-scroll\">Substitution Method<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Elimination_Method\" class=\"smooth-scroll\">Elimination Method<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Review\" class=\"smooth-scroll\">Review<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Linear_System_Practice_Questions\" class=\"smooth-scroll\">Linear System Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on using substitution and elimination to solve linear systems!<\/p>\n<p>\u201cSolving\u201d a system of equations means to determine the exact \\((x,y)\\) coordinate that satisfies both of the equations in the system. The process of solving a system depends on the structure of the equations. Some systems can easily be solved by <strong>graphing<\/strong> both equations and determining the exact point of <strong>intersection<\/strong>, while other systems are more suitable to be solved <strong<algebraically<\/strong>.<\/p>\n<p>This video will focus on two of the algebraic approaches, namely <strong>substitution<\/strong> and <strong>elimination<\/strong>. <\/p>\n<p>To solve a system of two equations with two <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/identifying-variables\/\">variables<\/a>, we are trying to find the exact point that satisfies both equations.  What do I mean by \u201csatisfy\u201d?<\/p>\n<p>That means that if you substitute that point into <em>BOTH<\/em> equations, you will have a \u201ctrue\u201d statement.<\/p>\n<p>We also already mentioned that the \u201cstructure\u201d of the system will determine which of the solving methods to use. <\/p>\n<h2><span id=\"Substitution_Method\" class=\"m-toc-anchor\"><\/span>Substitution Method<\/h2>\n<p>\nThe substitution method is typically used when one of the equations in the system is already solved for one of the variables. When this is the case, you can substitute the expression that the variable is equal to into the other equation:<\/p>\n<p>For example:<\/p>\n<div class=\"examplesentence\">\\(y=x+2\\)<br \/>\nand<br \/>\n\\(x+2y= -5\\)<\/div>\n<p>\n&nbsp;<br \/>\nBecause we know that the variable \\(y=x+2\\), we can substitute \\((x+2)\\) for the variable \\(y\\) in the second equation: <\/p>\n<div class=\"examplesentence\">\\(x+2(x+2)=-5\\)<\/div>\n<p>\n&nbsp;<br \/>\nNote that after the substitution, the \\(y\\)-variable is no longer in the equation.  <\/p>\n<p>At this point, we solve for \\(x\\) by distributing the coefficient of 2 into the parentheses.<\/p>\n<div class=\"examplesentence\">\\(x+2x+4= -5\\)<\/div>\n<p>\n&nbsp;<br \/>\nCombine like terms, \\(x+2x\\), and gather the constants by subtracting 4 from both sides of the equation:<\/p>\n<div class=\"examplesentence\">\\(x+2x+4= -5\\)<br \/>\n\\(3x+4-4= -5-4\\)<br \/>\n\\(3x= -9\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen we\u2019re going to divide both sides by 3 and this results in \\(x=-3\\).  <\/p>\n<p>At this point, we are halfway to our solution. In order to determine the \\(y\\)-variable, simply substitute \\(x= -3\\) into either of the original equations. Substituting into the first equation gives the quickest solution:<\/p>\n<div class=\"examplesentence\">\\(y=(-3)+2= -1\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe solution for this system is the <strong>ordered pair<\/strong> \\((-3,-1)\\). <\/p>\n<p>Let\u2019s practice the substitution method by looking at one more problem together:<\/p>\n<div class=\"examplesentence\">\\(y=-2x+4\\)<br \/>\n\\(3x+2y=1\\)<\/div>\n<p>\n&nbsp;<br \/>\nBecause \\(y\\) is solved in terms of \\(x\\) in the first equation, substitute the expression \\((-2x+4)\\) for \\(y\\) in the second equation.<\/p>\n<div class=\"examplesentence\">\\(3x+2(-2x+4)=1\\)<\/div>\n<p>\n&nbsp;<br \/>\nDistribute the 2 into the parentheses. Combine like terms and solve for \\(x\\):<\/p>\n<div class=\"examplesentence\">\\(3x-4x+8=1\\)<br \/>\n\\(-x+8=1\\)<br \/>\n\\(-x+8-8=1-8\\)<br \/>\n\\(-x=-7\\)<br \/>\n\\(x=7\\)<\/div>\n<p>\n&nbsp;<br \/>\nSolve for \\(y\\) by substituting \\(x= 7\\) into the first equation of the system:<\/p>\n<div class=\"examplesentence\">\\(y=-2(7)+4\\)<br \/>\n\\(y= -14+4\\)<br \/>\n\\(y= -10\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe ordered pair, \\((7,-10)\\), is the solution to this system.<\/p>\n<p>To check your work, simply substitute this <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/cartesian-coordinate-plane-and-graphing\/\">ordered<\/a> pair into both equations to verify that you get a true statement.<\/p>\n<p>Remember, these types of systems were well suited for the substitution method because one of the variables was already solved. <\/p>\n<h2><span id=\"Elimination_Method\" class=\"m-toc-anchor\"><\/span>Elimination Method<\/h2>\n<p>\nFor systems of equations that have two equations in \u201cstandard form,\u201d the elimination method may be preferred. Recall that standard-form equations have both variable terms on one side of the equation and constant terms on the other.<\/p>\n<p>The overall concept of this method is to \u201celiminate\u201d one of the variables from the system by adding like terms of each equation together. A straightforward example of the elimination method is shown:<\/p>\n<div class=\"examplesentence\">\\(2x+y=1\\)<br \/>\n\\(x -y=8\\)<\/div>\n<p>\n&nbsp;<br \/>\nBoth of these equations are in standard form. If the equations are added by combining like terms, the \\(y\\)-variable is eliminated because the leading coefficients of the \\(y\\)-variables are of opposite signs:<\/p>\n<div class=\"examplesentence\">\\(3x+0y=9\\)<br \/>\n\\(3x=9\\)<\/div>\n<p>\n&nbsp;<br \/>\nSolve for \\(x\\) by dividing both sides by 3, which results in \\(x=3\\). Now, as before, use this value to determine the value of \\(y\\) by substituting \\(x=3\\) into either of the original equations. Let\u2019s use the second equation of the system because it is fairly simple:<\/p>\n<div class=\"examplesentence\">\\(3-y=8\\)<\/div>\n<p>\n&nbsp;<br \/>\nTo solve for \\(y\\), subtract 3 from both sides and divide by -1 for a solution of \\(y=-5\\). So the final solution to this system is the ordered pair \\((3,-5)\\).<\/p>\n<p>As mentioned, that example was fairly straightforward because the coefficients of the \\(y\\)-variables canceled out when added. When coefficients do not cancel, you can use the <strong>Multiplication Property of Equality<\/strong> to multiply the equation by whatever value you need to eliminate.<\/p>\n<hr>\n<h2><span id=\"Review\" class=\"m-toc-anchor\"><\/span>Review<\/h2>\n<p>\nTo wrap up, let\u2019s see if you remember which method of solving would be most appropriate for the following systems:<\/p>\n<p>1. Which method of solving would be most appropriate for the following system?<\/p>\n<div class=\"examplesentence\">\\(-3x \u2212 3y = 3\\)<br \/>\n\\(y = \u22125x \u2212 17\\)<\/div>\n<p>\n&nbsp;<\/p>\n<div style=\"text-align: center; margin-bottom: 20px;\"><button class=\"buttontranscript\" onClick=\"toggle('Answer1')\">Show Answer<\/button><\/div>\n<div id=\"Answer1\" style=\"display:none; box-shadow: 1.5px 1.5px 5px grey; background-color:#E0E0E0; padding: 30px; padding-bottom: 15px; width: 60%; margin: auto; text-align: center;\">\n<strong>The correct answer is substitution.<\/strong><\/p>\n<p style=\"text-align: left;\">For this one, you would use substitution, since \\(y\\) is solved in the second equation.<\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\n2. Which method of solving would be most appropriate for the following system?<\/p>\n<div class=\"examplesentence\">\\(\u22124x + y = 6\\)<br \/>\n\\(\u22125x \u2212 y = 21\\)<\/div>\n<p>\n&nbsp;<\/p>\n<div style=\"text-align: center; margin-bottom: 20px;\"><button class=\"buttontranscript\" onClick=\"toggle('Answer2')\">Show Answer<\/button><\/div>\n<div id=\"Answer2\" style=\"display:none; box-shadow: 1.5px 1.5px 5px grey; background-color:#E0E0E0; padding: 30px; padding-bottom: 15px; width: 60%; margin: auto; text-align: center;\">\n<strong>The correct answer is elimination.<\/strong><\/p>\n<p style=\"text-align: left;\">For the second one, we\u2019d use elimination, since both equations are in the standard form and the \\(y\\)-term is eliminated by adding.<\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\nThanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Linear_System_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Linear System Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nSolve the following system of equations using substitution. <\/p>\n<div class=\"yellow-math-quote\">\n\\(x-2y=8\\)<br \/>\n\\(x+y=5\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\((-6, 1)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">\\((6, -1)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\((8, 5)\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\((-5, 8)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Start by solving the second equation for \\(x\\).<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(x+y=5\\)<br \/>\n\\(x=5-y\\)\n<\/p>\n<p>Now that we have isolated the variable \\(x\\), let\u2019s substitute this in for the \\(x\\) in the other equation.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(x-2y=8\\)<br \/>\n\\((5-y)-2y=8\\)\n<\/p>\n<p>From here we can solve for \\(y\\).<\/p>\n<p style=\"text-align: center\">\\(y=-1\\)<\/p>\n<p>Now let\u2019s take this value for \\(y\\) and plug it in to one of our original equations. Let\u2019s use the second original equation.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(x+y=5\\)<br \/>\n\\(x+(-1)=5\\)<\/p>\n<p>When we solve for \\(x\\), we have \\(x=6\\).<\/p>\n<p>Therefore, the solution is the ordered pair \\((6, -1)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nSolve the system of equations using elimination.<\/p>\n<div class=\"yellow-math-quote\">\n\\(12x-9y=37\\)<br \/>\n\\(8x+9y=23\\)\n<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">\\((3, &#8211;\\)<span style=\"font-size: 120%\">\\(\\frac{1}{9})\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\((4, -9)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\((\\)<span style=\"font-size: 120%\">\\(\\frac{1}{9}\\)<\/span>\\(,\\)<span style=\"font-size: 120%\">\\( \\frac{2}{9}\\)<\/span>\\()\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\((-3, &#8211;\\)<span style=\"font-size: 120%\">\\(\\frac{1}{9}\\)<\/span>\\()\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Elimination is a great choice for this problem because we can see that \\(-9y\\) and \\(9y\\) will cancel out nicely.<\/p>\n<p>Let\u2019s start by adding the two equations: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Complex-addition-problem.svg\" alt=\"Two linear equations, 12x - 9y = 37 and 8x + 9y = 23, are added to give 20x = 60, leading to the solution x = 3.\" width=\"173\" height=\"121\" class=\"aligncenter size-full wp-image-274165\"  role=\"img\" \/><\/p>\n<p>Now that we know \\(x=3\\), we can plug this in to one of our original equations in order to solve for \\(y\\).<\/p>\n<p>Let\u2019s use the second equation. <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(8x+9y=23\\)<br \/>\n\\(8(3)+9y=23\\)<\/p>\n<p>When we isolate the variable, we see that \\(y=-\\frac{1}{9}\\).<\/p>\n<p>Therefore, our solution is the ordered pair \\((3, -\\frac{1}{9})\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nSolve the system of equations using substitution.<\/p>\n<div class=\"yellow-math-quote\">\n\\(3x-y=7\\)<br \/>\n\\(y-2=2x\\)\n<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\((3, 2)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\((7, 20)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\((20, 9)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">\\((9, 20)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Start by solving the second equation for \\(y\\):<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(y-2=2x\\)<br \/>\n\\(y=2x+2\\)<\/p>\n<p>Now that we have solved for \\(y\\), we can substitute this into the other equation.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(3x-y=7\\)<br \/>\n\\(3x-(2x+2)=7\\)<\/p>\n<p>When we solve for \\(x\\), we have \\(x=9\\)<\/p>\n<p>From here, we can plug 9 in for \\(x\\) in one of the original equations in order to solve for \\(y\\).<\/p>\n<p>Let\u2019s use the second equation.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(y-2=2x\\)<br \/>\n\\(y-2=2(9)\\)<\/p>\n<p>When simplified, \\(y=20\\).<\/p>\n<p>Therefore, our solution is the ordered pair \\((9, 20)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFarmer Frank operates a farm with chickens and pigs. The total number of animals on the farm is 100. The total number of animal legs is 270. How many pigs are on the farm? How many chickens are on the farm?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">50 pigs and 50 chickens <\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">35 pigs and 65 chickens <\/div><div class=\"PQ\"  id=\"PQ-4-3\">25 pigs and 75 chickens <\/div><div class=\"PQ\"  id=\"PQ-4-4\">40 pigs and 60 chickens <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>We can solve this problem by setting up a system of equations.<\/p>\n<p>Let\u2019s write two equations that are true for the scenario. One equation is based on the number of animals and the other equation is based on the number of legs, where \\(c\\) represents chickens and \\(p\\) represents pigs.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(c+p=100\\)<br \/>\n\\(2c+4p=270\\)<\/p>\n<p>From here, we can solve the first equation for \\(p\\). <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(c+p=100\\)<br \/>\n\\(p=100-c\\)<\/p>\n<p>Now we plug this in for \\(p\\) in the other equation. <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(2c+4p=270\\)<br \/>\n\\(2c+4(100-c)=270\\)<\/p>\n<p>Now solve for \\(c\\): <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(2c+4(100-c)=270\\)<br \/>\n\\(c=65\\)<\/p>\n<p>From here ,we can plug in the value of \\(c\\) into one of our original equations in order to solve for \\(p\\). <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(c+p=100\\)<br \/>\n\\((65)+p=100\\)<\/p>\n<p>Therefore, \\(p=35\\), which means Farmer Frank has 35 pigs and 65 chickens on his farm.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMr. Parker wanted to treat his family so he took them all out to a movie. There are 15 people in his family including himself. The adult tickets were $10 and the children tickets were $5. The total ticket cost was $110. How many children and how many adults are in the family? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">4 children and 11 adults<\/div><div class=\"PQ\"  id=\"PQ-5-2\">3 children and 12 adults <\/div><div class=\"PQ\"  id=\"PQ-5-3\">5 children and 10 adults <\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">8 children and 7 adults <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Start by setting up two true equations that represent the scenario. One equation can represent the number of people and the other can represent cost, where \\(a\\) will represent adults and \\(c\\) will represent children. <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(a+c=15\\)<br \/>\n\\(10a+5c=110\\)<\/p>\n<p>Let\u2019s solve the first equation for \\(a\\). <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(a+c=15\\)<br \/>\n\\(a=15\u2013c\\)<\/p>\n<p>Now let\u2019s substitute this in for \\(a\\) in the other equation. <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(10a+5c=110\\)<br \/>\n\\(10(15 -c)+5c=110\\)<br \/>\n\\(c=8\\)<\/p>\n<p>Now that we know \\(c=8\\), we can plug this into one of the original equations in order to solve for \\(a\\).<\/p>\n<p>Let\u2019s use the first equation: <\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(a+c=15\\)<br \/>\n\\(a+(8)=15\\)<br \/>\n\\(a=7\\)<\/p>\n<p>Therefore, there are 8 children and 7 adults in the family.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n<p><script>\nfunction toggle(obj) {\n          var obj=document.getElementById(obj);\n          if (obj.style.display == \"block\") obj.style.display = \"none\";\n          else obj.style.display = \"block\";\n}\n<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":187067,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-4347","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-systems-of-equations-and-their-solutions-videos","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"content_type-practice-questions","11":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4347","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=4347"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4347\/revisions"}],"predecessor-version":[{"id":280019,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4347\/revisions\/280019"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/187067"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=4347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}