{"id":15364,"date":"2021-10-11T23:05:38","date_gmt":"2021-10-11T23:05:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/3-2-2-the-elimination-method\/"},"modified":"2021-10-16T20:45:37","modified_gmt":"2021-10-16T20:45:37","slug":"3-2-2-the-elimination-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/3-2-2-the-elimination-method\/","title":{"raw":"The Elimination Method Without Multiplication","rendered":"The Elimination Method Without Multiplication"},"content":{"raw":"\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Use the elimination method without multiplication<\/li>\n \t<li>Express the solution of an inconsistent system of equations containing two variables<\/li>\n \t<li>Express the solution of a dependent system of equations containing two variables<\/li>\n<\/ul>\n<\/div>\nThe <b>elimination method<\/b> for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation to eliminate one of the variable terms. &nbsp;In this method, you may or may not need to multiply the terms in one equation by a number first. &nbsp;We will first look at examples where no multiplication is necessary to use the elimination method. &nbsp;In the next section you will see examples using multiplication after you are familiar with the idea of the elimination method.\n\nIt is easier to show rather than tell with this method, so let's dive right into some examples.\n\nIf you add the two equations,\n\n[latex]x\u2013y=\u22126[\/latex] and [latex]x+y=8[\/latex] together, watch what happens.\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,\\,\\,x-y=\\,-6\\\\\\underline{+\\,x+y=\\,\\,\\,8}\\\\\\,2x+0\\,=\\,\\,\\,\\,2\\end{array}[\/latex]<\/p>\nYou have eliminated the y term, and this equation can be solved using the methods for solving equations with one variable.\n\nLet\u2019s see how this system is solved using the elimination method.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nUse elimination to solve the system.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=\u22126\\\\x+y=\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"403819\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"403819\"]Add the equations.\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-y=\\,\\,-6\\\\+\\underline{\\,\\,x+y=\\,\\,\\,\\,\\,8}\\\\\\,\\,\\,\\,\\,\\,2x\\,\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\,2\\end{array}[\/latex]<\/p>\nSolve for [latex]x[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x=2\\\\x=1\\end{array}[\/latex]<\/p>\nSubstitute [latex]x=1[\/latex] into one of the original equations and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x+y=8\\\\1+y=8\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=8\u20131\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=7\\end{array}[\/latex]<\/p>\nBe sure to check your answer in both equations!\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=\u22126\\\\1\u20137=\u22126\\\\\u22126=\u22126\\\\\\text{TRUE}\\\\\\\\x+y=8\\\\1+7=8\\\\8=8\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\nThe answers check.\n<h4>Answer<\/h4>\nThe solution is [latex](1, 7)[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\nLet's look an another example of solving by elimination that works out nicely.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nSolve the given system of equations by elimination.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+2y=-1\\hfill \\\\ -x+y=3\\hfill \\end{array}[\/latex]<\/div>\n[reveal-answer q=\"522070\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"522070\"]\n\nBoth equations are already set equal to a constant. Notice that the coefficient of [latex]x[\/latex] in the second equation,&nbsp;[latex]\u20131[\/latex], is the opposite of the coefficient of [latex]x[\/latex] in the first equation,&nbsp;[latex]1[\/latex]. As a result, we can simply add the two equations together to eliminate [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{l}\\hfill \\\\ \\:\\:x+2y=-1\\hfill \\\\ -x+y=3\\hfill \\end{array}}{\\text{}\\text{}\\text{}\\text{}\\text{}\\:\\:\\:\\:\\:\\:3y=2}[\/latex]<\/div>\n&nbsp;\n\nNow that we have eliminated [latex]x[\/latex], we can solve the resulting equation for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3y=2\\hfill \\\\ \\text{ }y=\\dfrac{2}{3}\\hfill \\end{array}[\/latex]<\/p>\nThen, we substitute this value for [latex]y[\/latex] into one of the original equations and solve for [latex]x[\/latex]\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }-x+y=3\\hfill \\\\ \\text{ }-x+\\dfrac{2}{3}=3\\hfill \\\\ \\text{ }-x=3-\\dfrac{2}{3}\\hfill \\\\ \\text{ }-x=\\dfrac{7}{3}\\hfill \\\\ \\text{ }\\:\\:\\:\\:\\:x=-\\dfrac{7}{3}\\hfill \\end{array}[\/latex]<\/p>\nThe solution to this system is [latex]\\left(-\\dfrac{7}{3},\\dfrac{2}{3}\\right)[\/latex].\n\nCheck the solution in the first equation.\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}\\text{ }x+2y=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }\\left(-\\dfrac{7}{3}\\right)+2\\left(\\dfrac{2}{3}\\right)=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }-\\dfrac{7}{3}+\\dfrac{4}{3}=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }-\\dfrac{3}{3}=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }-1=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{True}\\hfill \\end{array}[\/latex]<\/p>\nWe gain an important perspective on systems of equations by looking at the graphical representation. In the graph below, you will see that the equations intersect at the solution. We do not need to ask whether there may be a second solution, because observing the graph confirms that the system has exactly one solution.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222638\/CNX_Precalc_Figure_09_01_0042.jpg\" alt=\"A graph of two lines that cross at the point negative seven-thirds, two-thirds. The first line's equation is x+2y=negative 1. The second line's equation is negative x + y equals 3.\" width=\"487\" height=\"291\">\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n\nUnfortunately not all systems work out this easily. How about a system like [latex]2x+y=12[\/latex] and [latex]\u22123x+y=2[\/latex]. If you add these two equations together, no variables are eliminated.\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,\\,2x+y=12\\\\\\underline{-3x+y=\\,\\,\\,2}\\\\-x+2y=14\\end{array}[\/latex]<\/p>\nBut you want to eliminate a variable. So let\u2019s add the opposite of one of the equations to the other equation. This means multiply every term in one of the equations by [latex]-1[\/latex], so that the sign of every terms is opposite.\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,2x+\\,\\,y\\,=12\\rightarrow2x+y=12\\rightarrow2x+y=12\\\\\u22123x+\\,\\,y\\,=2\\rightarrow\u2212\\left(\u22123x+y\\right)=\u2212(2)\\rightarrow3x\u2013y=\u22122\\\\\\,\\,\\,\\,5x+0y=10\\end{array}[\/latex]<\/p>\nYou have eliminated the y variable, and the problem can now be solved.\n\nThe following video describes a similar problem where you can eliminate one variable by adding the two equations together.\n\nhttps:\/\/youtu.be\/M4IEmwcqR3c\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"63\" height=\"56\">Caution! &nbsp;When you add the opposite of one entire equation to another, make sure to change the sign of EVERY term on both sides of the equation. This is a very common&nbsp;mistake to make.<\/div>\n&nbsp;\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nUse elimination to solve the system.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\\u22123x+y=2\\,\\,\\,\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"702178\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"702178\"]You can eliminate the y-variable if you add the opposite of one of the equations to the other equation.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\\u22123x+y=2\\,\\,\\,\\end{array}[\/latex]<\/p>\nRewrite the second equation as its opposite.\n\nAdd.&nbsp;Solve for [latex]x[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\,\\\\3x\u2013y=\u22122\\\\5x=10\\,\\\\x=2\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\nSubstitute [latex]y=2[\/latex] into one of the original equations and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\left(2\\right)+y=12\\\\4+y=12\\\\y=8\\,\\,\\,\\end{array}[\/latex]<\/p>\nBe sure to check your answer in both equations!\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\2\\left(2\\right)+8=12\\\\4+8=12\\\\12=12\\\\\\text{TRUE}\\\\\\\\\u22123x+y=2\\\\\u22123\\left(2\\right)+8=2\\\\\u22126+8=2\\\\2=2\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\nThe answers check.\n<h4>Answer<\/h4>\nThe solution is [latex](2, 8)[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]38342[\/ohm_question]\n\n<\/div>\nThe following are two more examples showing how to solve linear systems of equations using elimination.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nUse elimination to solve the system.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=\\,25\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"438400\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"438400\"]Notice the coefficients of each variable in each equation. If you add these two equations, the x term will be eliminated since [latex]\u22122x+2x=0[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=\\,25\\end{array}[\/latex]<\/p>\nAdd and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=25\\,\\\\8y=24\\,\\\\y=3\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\nSubstitute [latex]y=3[\/latex] into one of the original equations.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+5y=25\\\\2x+5\\left(3\\right)=25\\\\2x+15=25\\\\2x=10\\\\x=5\\,\\,\\,\\end{array}[\/latex]<\/p>\nCheck solutions.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\\u22122\\left(5\\right)+3\\left(3\\right)=\u22121\\\\\u221210+9=\u22121\\\\\u22121=\u22121\\\\\\text{TRUE}\\\\\\\\2x+5y=25\\\\2\\left(5\\right)+5\\left(3\\right)=25\\\\10+15=25\\\\25=25\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\nThe answers check.\n<h4>Answer<\/h4>\nThe solution is [latex](5, 3)[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nUse elimination to solve for [latex]x[\/latex] and [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\5x+2y=16\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"776093\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"776093\"]Notice the coefficients of each variable in each equation. You will need to add the opposite of one of the equations to eliminate the variable y, as [latex]2y+2y=4y[\/latex], but [latex]2y+\\left(\u22122y\\right)=0[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\5x+2y=16\\end{array}[\/latex]<\/p>\n&nbsp;Change one of the equations to its opposite, add and solve for [latex]x[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\,\\,\\,\\,\\\\\u22125x\u20132y=\u221216\\\\\u2212x=\u22122\\,\\,\\,\\\\x=2\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\nSubstitute [latex]x=2[\/latex] into one of the original equations and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\4\\left(2\\right)+2y=14\\\\8+2y=14\\\\2y=6\\,\\,\\,\\\\y=3\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<h4>Answer<\/h4>\nThe solution is [latex](2, 3)[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\nGo ahead and check this last example\u2014substitute [latex](2, 3)[\/latex] into both equations. You get two true statements: [latex]14=14[\/latex] and [latex]16=16[\/latex]!\n\nNotice that you could have used the opposite of the first equation rather than the second equation and gotten the same result.\n<h2 id=\"video1\" class=\"no-indent\" style=\"text-align: left;\">Recognize systems that have no solution or an infinite number of solutions<\/h2>\nJust as with the substitution method, the elimination method will sometimes eliminate both variables, and you end up with either a true statement or a false statement. Recall that a false statement means that there is no solution.\n\nLet\u2019s look at an example.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nSolve for [latex]x[\/latex] and [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-x\u2013y=-4\\\\x+y=2\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"101540\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"101540\"]Add the equations to eliminate the&nbsp;[latex]x[\/latex]-term.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-x\u2013y=-4\\\\\\underline{x+y=2\\,\\,\\,}\\\\0=\u22122\\end{array}[\/latex]<\/p>\n\n<h4>Answer<\/h4>\nThere is no solution.\n\n[\/hidden-answer]\n\n<\/div>\nGraphing these lines shows that they are parallel lines and as such do not share any point in common, verifying that there is no solution.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064435\/image040-1.jpg\" alt=\"Two parallel lines. One line is -x-y=-4. The other line is x+y=2.\" width=\"346\" height=\"345\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]115196[\/ohm_question]\n\n<\/div>\nIf both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. In fact, the equations are the same line.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nSolve for [latex]x[\/latex] and [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+y=2\\,\\,\\,\\,\\\\-x\u2212y=-2\\end{array}[\/latex]<\/p>\n[reveal-answer q=\"328100\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"328100\"]Add the equations to eliminate the&nbsp;x-term.\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+y=2\\,\\,\\,\\,\\\\\\underline{-x\u2212y=-2}\\\\0=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n\n<h4>Answer<\/h4>\nThere are an infinite number of solutions.\n\n[\/hidden-answer]\n\n<\/div>\nGraphing these two equations will help to illustrate what is happening.\n\n<b><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064437\/image041-1.jpg\" alt=\"Two overlapping lines. One is -x-y=-2, and the other is x+y=2.\" width=\"346\" height=\"345\"><\/b>\n<h2 id=\"video3\" class=\"no-indent\" style=\"text-align: left;\"><\/h2>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]115192[\/ohm_question]\n\n<\/div>\n&nbsp;\n\nIn the following video, a system of equations which has no solutions is solved using the method of elimination.\n\nhttps:\/\/youtu.be\/z5_ACYtzW98\n<h2>Summary<\/h2>\nCombining equations is a powerful tool for solving a system of equations. Adding or subtracting two equations in order to eliminate a common variable is called the elimination (or addition) method. Once one variable is eliminated, it becomes much easier to solve for the other one.\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1IAdNoHAjCf8mcHfCrRLdq1rt-2dHcq0B65sVWobuIsA\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the elimination method without multiplication<\/li>\n<li>Express the solution of an inconsistent system of equations containing two variables<\/li>\n<li>Express the solution of a dependent system of equations containing two variables<\/li>\n<\/ul>\n<\/div>\n<p>The <b>elimination method<\/b> for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation to eliminate one of the variable terms. &nbsp;In this method, you may or may not need to multiply the terms in one equation by a number first. &nbsp;We will first look at examples where no multiplication is necessary to use the elimination method. &nbsp;In the next section you will see examples using multiplication after you are familiar with the idea of the elimination method.<\/p>\n<p>It is easier to show rather than tell with this method, so let&#8217;s dive right into some examples.<\/p>\n<p>If you add the two equations,<\/p>\n<p>[latex]x\u2013y=\u22126[\/latex] and [latex]x+y=8[\/latex] together, watch what happens.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,\\,\\,x-y=\\,-6\\\\\\underline{+\\,x+y=\\,\\,\\,8}\\\\\\,2x+0\\,=\\,\\,\\,\\,2\\end{array}[\/latex]<\/p>\n<p>You have eliminated the y term, and this equation can be solved using the methods for solving equations with one variable.<\/p>\n<p>Let\u2019s see how this system is solved using the elimination method.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use elimination to solve the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=\u22126\\\\x+y=\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q403819\">Show Solution<\/span><\/p>\n<div id=\"q403819\" class=\"hidden-answer\" style=\"display: none\">Add the equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-y=\\,\\,-6\\\\+\\underline{\\,\\,x+y=\\,\\,\\,\\,\\,8}\\\\\\,\\,\\,\\,\\,\\,2x\\,\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\,2\\end{array}[\/latex]<\/p>\n<p>Solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x=2\\\\x=1\\end{array}[\/latex]<\/p>\n<p>Substitute [latex]x=1[\/latex] into one of the original equations and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x+y=8\\\\1+y=8\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=8\u20131\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=7\\end{array}[\/latex]<\/p>\n<p>Be sure to check your answer in both equations!<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\u2013y=\u22126\\\\1\u20137=\u22126\\\\\u22126=\u22126\\\\\\text{TRUE}\\\\\\\\x+y=8\\\\1+7=8\\\\8=8\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>The answers check.<\/p>\n<h4>Answer<\/h4>\n<p>The solution is [latex](1, 7)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Let&#8217;s look an another example of solving by elimination that works out nicely.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the given system of equations by elimination.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+2y=-1\\hfill \\\\ -x+y=3\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q522070\">Show Solution<\/span><\/p>\n<div id=\"q522070\" class=\"hidden-answer\" style=\"display: none\">\n<p>Both equations are already set equal to a constant. Notice that the coefficient of [latex]x[\/latex] in the second equation,&nbsp;[latex]\u20131[\/latex], is the opposite of the coefficient of [latex]x[\/latex] in the first equation,&nbsp;[latex]1[\/latex]. As a result, we can simply add the two equations together to eliminate [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{l}\\hfill \\\\ \\:\\:x+2y=-1\\hfill \\\\ -x+y=3\\hfill \\end{array}}{\\text{}\\text{}\\text{}\\text{}\\text{}\\:\\:\\:\\:\\:\\:3y=2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Now that we have eliminated [latex]x[\/latex], we can solve the resulting equation for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3y=2\\hfill \\\\ \\text{ }y=\\dfrac{2}{3}\\hfill \\end{array}[\/latex]<\/p>\n<p>Then, we substitute this value for [latex]y[\/latex] into one of the original equations and solve for [latex]x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }-x+y=3\\hfill \\\\ \\text{ }-x+\\dfrac{2}{3}=3\\hfill \\\\ \\text{ }-x=3-\\dfrac{2}{3}\\hfill \\\\ \\text{ }-x=\\dfrac{7}{3}\\hfill \\\\ \\text{ }\\:\\:\\:\\:\\:x=-\\dfrac{7}{3}\\hfill \\end{array}[\/latex]<\/p>\n<p>The solution to this system is [latex]\\left(-\\dfrac{7}{3},\\dfrac{2}{3}\\right)[\/latex].<\/p>\n<p>Check the solution in the first equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}\\text{ }x+2y=-1\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }\\left(-\\dfrac{7}{3}\\right)+2\\left(\\dfrac{2}{3}\\right)=-1\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }-\\dfrac{7}{3}+\\dfrac{4}{3}=-1\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }-\\dfrac{3}{3}=-1\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }-1=-1\\hfill & \\hfill & \\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\n<p>We gain an important perspective on systems of equations by looking at the graphical representation. In the graph below, you will see that the equations intersect at the solution. We do not need to ask whether there may be a second solution, because observing the graph confirms that the system has exactly one solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222638\/CNX_Precalc_Figure_09_01_0042.jpg\" alt=\"A graph of two lines that cross at the point negative seven-thirds, two-thirds. The first line's equation is x+2y=negative 1. The second line's equation is negative x + y equals 3.\" width=\"487\" height=\"291\" \/>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Unfortunately not all systems work out this easily. How about a system like [latex]2x+y=12[\/latex] and [latex]\u22123x+y=2[\/latex]. If you add these two equations together, no variables are eliminated.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,\\,2x+y=12\\\\\\underline{-3x+y=\\,\\,\\,2}\\\\-x+2y=14\\end{array}[\/latex]<\/p>\n<p>But you want to eliminate a variable. So let\u2019s add the opposite of one of the equations to the other equation. This means multiply every term in one of the equations by [latex]-1[\/latex], so that the sign of every terms is opposite.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,2x+\\,\\,y\\,=12\\rightarrow2x+y=12\\rightarrow2x+y=12\\\\\u22123x+\\,\\,y\\,=2\\rightarrow\u2212\\left(\u22123x+y\\right)=\u2212(2)\\rightarrow3x\u2013y=\u22122\\\\\\,\\,\\,\\,5x+0y=10\\end{array}[\/latex]<\/p>\n<p>You have eliminated the y variable, and the problem can now be solved.<\/p>\n<p>The following video describes a similar problem where you can eliminate one variable by adding the two equations together.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Solve a System of Equations Using the Elimination Method\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/M4IEmwcqR3c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"63\" height=\"56\" \/>Caution! &nbsp;When you add the opposite of one entire equation to another, make sure to change the sign of EVERY term on both sides of the equation. This is a very common&nbsp;mistake to make.<\/div>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use elimination to solve the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\\u22123x+y=2\\,\\,\\,\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q702178\">Show Solution<\/span><\/p>\n<div id=\"q702178\" class=\"hidden-answer\" style=\"display: none\">You can eliminate the y-variable if you add the opposite of one of the equations to the other equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\\u22123x+y=2\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Rewrite the second equation as its opposite.<\/p>\n<p>Add.&nbsp;Solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\,\\\\3x\u2013y=\u22122\\\\5x=10\\,\\\\x=2\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Substitute [latex]y=2[\/latex] into one of the original equations and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\left(2\\right)+y=12\\\\4+y=12\\\\y=8\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Be sure to check your answer in both equations!<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+y=12\\\\2\\left(2\\right)+8=12\\\\4+8=12\\\\12=12\\\\\\text{TRUE}\\\\\\\\\u22123x+y=2\\\\\u22123\\left(2\\right)+8=2\\\\\u22126+8=2\\\\2=2\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>The answers check.<\/p>\n<h4>Answer<\/h4>\n<p>The solution is [latex](2, 8)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm38342\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=38342&theme=oea&iframe_resize_id=ohm38342&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following are two more examples showing how to solve linear systems of equations using elimination.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use elimination to solve the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=\\,25\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q438400\">Show Solution<\/span><\/p>\n<div id=\"q438400\" class=\"hidden-answer\" style=\"display: none\">Notice the coefficients of each variable in each equation. If you add these two equations, the x term will be eliminated since [latex]\u22122x+2x=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=\\,25\\end{array}[\/latex]<\/p>\n<p>Add and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\2x+5y=25\\,\\\\8y=24\\,\\\\y=3\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Substitute [latex]y=3[\/latex] into one of the original equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x+5y=25\\\\2x+5\\left(3\\right)=25\\\\2x+15=25\\\\2x=10\\\\x=5\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Check solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22122x+3y=\u22121\\\\\u22122\\left(5\\right)+3\\left(3\\right)=\u22121\\\\\u221210+9=\u22121\\\\\u22121=\u22121\\\\\\text{TRUE}\\\\\\\\2x+5y=25\\\\2\\left(5\\right)+5\\left(3\\right)=25\\\\10+15=25\\\\25=25\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>The answers check.<\/p>\n<h4>Answer<\/h4>\n<p>The solution is [latex](5, 3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use elimination to solve for [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\5x+2y=16\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q776093\">Show Solution<\/span><\/p>\n<div id=\"q776093\" class=\"hidden-answer\" style=\"display: none\">Notice the coefficients of each variable in each equation. You will need to add the opposite of one of the equations to eliminate the variable y, as [latex]2y+2y=4y[\/latex], but [latex]2y+\\left(\u22122y\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\5x+2y=16\\end{array}[\/latex]<\/p>\n<p>&nbsp;Change one of the equations to its opposite, add and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\,\\,\\,\\,\\\\\u22125x\u20132y=\u221216\\\\\u2212x=\u22122\\,\\,\\,\\\\x=2\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Substitute [latex]x=2[\/latex] into one of the original equations and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x+2y=14\\\\4\\left(2\\right)+2y=14\\\\8+2y=14\\\\2y=6\\,\\,\\,\\\\y=3\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The solution is [latex](2, 3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Go ahead and check this last example\u2014substitute [latex](2, 3)[\/latex] into both equations. You get two true statements: [latex]14=14[\/latex] and [latex]16=16[\/latex]!<\/p>\n<p>Notice that you could have used the opposite of the first equation rather than the second equation and gotten the same result.<\/p>\n<h2 id=\"video1\" class=\"no-indent\" style=\"text-align: left;\">Recognize systems that have no solution or an infinite number of solutions<\/h2>\n<p>Just as with the substitution method, the elimination method will sometimes eliminate both variables, and you end up with either a true statement or a false statement. Recall that a false statement means that there is no solution.<\/p>\n<p>Let\u2019s look at an example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-x\u2013y=-4\\\\x+y=2\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q101540\">Show Solution<\/span><\/p>\n<div id=\"q101540\" class=\"hidden-answer\" style=\"display: none\">Add the equations to eliminate the&nbsp;[latex]x[\/latex]-term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-x\u2013y=-4\\\\\\underline{x+y=2\\,\\,\\,}\\\\0=\u22122\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Graphing these lines shows that they are parallel lines and as such do not share any point in common, verifying that there is no solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064435\/image040-1.jpg\" alt=\"Two parallel lines. One line is -x-y=-4. The other line is x+y=2.\" width=\"346\" height=\"345\" \/><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm115196\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=115196&theme=oea&iframe_resize_id=ohm115196&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>If both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. In fact, the equations are the same line.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+y=2\\,\\,\\,\\,\\\\-x\u2212y=-2\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q328100\">Show Solution<\/span><\/p>\n<div id=\"q328100\" class=\"hidden-answer\" style=\"display: none\">Add the equations to eliminate the&nbsp;x-term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+y=2\\,\\,\\,\\,\\\\\\underline{-x\u2212y=-2}\\\\0=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>There are an infinite number of solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Graphing these two equations will help to illustrate what is happening.<\/p>\n<p><b><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064437\/image041-1.jpg\" alt=\"Two overlapping lines. One is -x-y=-2, and the other is x+y=2.\" width=\"346\" height=\"345\" \/><\/b><\/p>\n<h2 id=\"video3\" class=\"no-indent\" style=\"text-align: left;\"><\/h2>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm115192\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=115192&theme=oea&iframe_resize_id=ohm115192&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video, a system of equations which has no solutions is solved using the method of elimination.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  System of Equations Using Elimination (No Solution)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z5_ACYtzW98?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Combining equations is a powerful tool for solving a system of equations. Adding or subtracting two equations in order to eliminate a common variable is called the elimination (or addition) method. Once one variable is eliminated, it becomes much easier to solve for the other one.<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1IAdNoHAjCf8mcHfCrRLdq1rt-2dHcq0B65sVWobuIsA\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15364\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Solve a System of Equations Using the Elimination Method . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/M4IEmwcqR3c\">https:\/\/youtu.be\/M4IEmwcqR3c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: Solve a System of Equations Using the Elimination Method. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_liDhKops2w\">https:\/\/youtu.be\/_liDhKops2w<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Ex: System of Equations Using Elimination (Infinite Solutions) . <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/NRxh9Q16Ulk\">https:\/\/youtu.be\/NRxh9Q16Ulk<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Solve a System of Equations Using the Elimination Method \",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/M4IEmwcqR3c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Solve a System of Equations Using the Elimination Method\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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