{"id":122,"date":"2023-11-08T16:09:58","date_gmt":"2023-11-08T16:09:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/solving-systems-of-three-equations-in-three-variables\/"},"modified":"2026-02-05T10:00:39","modified_gmt":"2026-02-05T10:00:39","slug":"2-2-systems-of-three-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/2-2-systems-of-three-linear-equations\/","title":{"raw":"2.2 Systems of Three Linear Equations","rendered":"2.2 Systems of Three Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether an ordered triple is a solution of a system of three linear equations with three variables.<\/li>\r\n \t<li>Solve systems of three linear equations with three variables.<\/li>\r\n \t<li>Determine if a system of three linear equations with three variables is inconsistent or consistent.<\/li>\r\n \t<li>Determine if equations are dependent or independent for a system of three linear equations with three variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\nSometimes in problems or applications there are three (or more) unknowns. A system with three unknowns requires three equations in order to generally have a single solution (although just like with two variables, there are special cases.) We typically use [latex]z[\/latex] for the third variable. An example of a system with three variables is shown here:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\phantom{6}x+\\phantom{4}y+\\phantom{5}z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/p>\r\nA <strong>solution <\/strong>to a system of three equations in three variables is an ordered triple [latex]\\left(x,y,z\\right)[\/latex] that makes all three equations true. Graphically, the solution set to a single linear equation in three variables can be represented by a plane in three dimensions. A plane is a flat two-dimensional surface that extends infinitely. A system of three equations represents three planes, and a solution represents a common intersection point.\r\n\r\n<img class=\"aligncenter wp-image-2389 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11211012\/Screen-Shot-2016-07-11-at-2.08.59-PM.png\" alt=\"Three planes intersecting at a point.\" width=\"262\" height=\"210\" \/>\r\n\r\nYou can visualize such an intersection by imagining any corner in a rectangular room. A line is defined by the the intersection of two planes: for example, two adjacent walls. A corner is defined by the intersection of three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.\r\n\r\nIn the first example, we will determine whether an ordered triple is a solution for a system of three linear equations in three variables.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.\r\n<div>[latex]\\begin{array}{l}\\phantom{6}x+\\phantom{4}y+\\phantom{5}z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"642719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"642719\"]\r\n\r\nWe will check each equation by substituting in the values of the ordered triple for [latex]x,y[\/latex], and [latex]z[\/latex].\r\n\r\n[latex]\\begin{array}{ccccc}\\begin{array}{r}\\hfill x+y+z=2\\\\ \\hfill \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\hfill \\text{True}\\end{array}&amp; &amp; \\begin{array}{r}\\hfill \\text{}6x - 4y+5z=31\\\\ \\hfill 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ \\hfill 18+8+5=31\\\\ \\hfill \\text{True}\\end{array}&amp; &amp; \\begin{array}{r}\\hfill \\text{}5x+2y+2z=13\\\\ \\hfill 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ \\hfill \\text{}15 - 4+2=13\\\\ \\hfill \\text{True}\\end{array}\\end{array}[\/latex]\r\n\r\nThe ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system. Note that if any of the three equations was not true, it would not be a solution to the system.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving a Three-Variable System<\/h2>\r\nAs with many math processes, our goal is to reduce our problem to a problem that was previously solved. If we can eliminate one of the variables, that will result in a system with only two variables and we know how to solve those.\r\n\r\nConsider the following example, which begins with [latex]x[\/latex] already removed from two of the equations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the system of equations.\r\n<div style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}x-\\phantom{32}y+\\phantom{2}z=5&amp;(1)\\\\\\phantom{x-}-2y+\\phantom{2}z=6&amp;(2)\\\\\\phantom{x-3}2y-\\phantom{2}2z=-12&amp;(3)\\end{array}\\right.[\/latex]<\/div>\r\n[reveal-answer q=\"223787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223787\"]\r\n\r\nIt is good practice to label the equations as we go because we will need to reference them later. The key observation is that equations [latex](2)[\/latex] and [latex](3)[\/latex] only have the variables [latex]y[\/latex] and [latex]z,[\/latex] so we can solve them as a two variable system and find the solution for\u00a0[latex]y[\/latex] and [latex]z[\/latex] immediately.\r\n<p style=\"text-align: center;\">[latex]\\left\\{ \\begin{array}{rl}-2y+z&amp;=\\text{ }6\\\\ 2y-2z&amp;=\\text{ }-12\\end{array}\\right.[\/latex]<\/p>\r\nAdding the equations right away will eliminate [latex]y[\/latex] and allow us to find the value of [latex]z[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rl}\\hfill \\\\ \\:\\:-2y+z&amp;=6\\hfill \\\\ 2y-2z&amp;=-12\\hfill \\end{array}}{\\text{}\\text{}\\text{}\\text{}\\text{}\\:\\:\\:\\:\\:\\:-z=-6}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]z=6[\/latex]<\/p>\r\nNow we can substitute the value for\u00a0[latex]z[\/latex] that we obtained into equation [latex](2)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl}-2y+(6)&amp;=\\;6\\\\-2y&amp;=\\;6-6\\\\-2y&amp;=\\;0\\\\\\,\\,\\,\\,y&amp;=\\;0\\end{array}[\/latex]<\/p>\r\nBe careful here not to get confused with a solution of\u00a0[latex]y = 0[\/latex] and an inconsistent solution. It is ok for variables to equal\u00a0[latex]0.[\/latex]\r\n\r\nNow we can substitute\u00a0[latex]z = 6[\/latex] and\u00a0[latex]y = 0[\/latex] back into equation [latex](1)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl}x-y+z&amp;=\\;5\\\\x-0+6&amp;=\\;5\\\\x+6&amp;=\\;5\\\\x&amp;=\\;5-6\\\\x&amp;=\\;-1\\end{array}[\/latex]<\/p>\r\nThe solution is the ordered triple [latex](x,y,z)=(-1,0,6).[\/latex] We leave it to you to check that this solution satisfies all three equations.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe strategy we used in the preceding example informs the general process that we will use to solve systems with three unknowns.\r\n<div class=\"textbox shaded\">\r\n<h3>Solving a linear system with three unknowns<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Pick any pair of equations and eliminate one variable.<\/li>\r\n \t<li>Pick another pair of equations and eliminate the same variable.<\/li>\r\n \t<li>You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.<\/li>\r\n \t<li>Back-substitute known variables into any one of the original equations and solve for the missing variable.<\/li>\r\n<\/ol>\r\n<\/div>\r\nSince our previous example started already in Step 3, we will now do an example where we must do all four of the steps.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve the system of equations.\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}2x+\\phantom{4}y+4z=3&amp;(1)\\\\ -x + 3y-4z=6&amp;(2)\\\\ 3x-5y+4z=2&amp;(3) \\end{array}\\right.[\/latex]<\/p>\r\n[reveal-answer q=\"535211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"535211\"]\r\n\r\nFirst we should identify which variable we plan to eliminate in Steps 1 and 2. Notice that the coefficients for [latex]z[\/latex] are all either [latex]4[\/latex] or\u00a0[latex]-4.[\/latex] This suggests eliminating\u00a0[latex]z[\/latex] in two pairs of equations.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rll}2x+\\phantom{4}y+4z=3&amp;(1) \\\\ -x + 3y-4z=6&amp;(2) \\end{array}}{\\phantom{2}x+4y\\phantom{-24z}=9 \\:\\:\\:\\:(4)}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rll}-x+3y-4z=6&amp;(2) \\\\ 3x -5y+4z=2&amp;(3) \\end{array}}{2x -2y\\phantom{-24z}=8 \\:\\:\\:\\:(5)}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Equations [latex](4)[\/latex] and\u00a0[latex](5)[\/latex] form our two variable system,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left\\{ \\begin{array}{rll}x+4y&amp;=\\:9&amp;(4)\\\\ 2x-2y&amp;=\\:8&amp;(5)\\end{array}\\right.[\/latex]<\/p>\r\nWe now solve this two variable system by elimination.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl}x+4y&amp;=9&amp;\\xrightarrow{\\cdot \\; -2} &amp; -2x-8y &amp;=-18\\\\\r\n2x-2y&amp;=8&amp;\\xrightarrow{\\phantom{\\cdot \\; -2}} &amp; 2x-2y&amp;=8 \\\\\r\n&amp; &amp; &amp; \\text{_______}&amp;\\text{______} \\\\\r\n&amp; &amp; &amp; -10y&amp;=-10 \\\\\r\n&amp; &amp; &amp; y &amp;=1 \\\\ \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}x+4y&amp;=9&amp;(4)\\\\\r\nx+4(1)&amp;=9&amp; \\\\\r\nx&amp;=5 &amp; \\end{array}[\/latex]<\/p>\r\nNow, substitute [latex]x=5, y=1[\/latex] into equation [latex](1).[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}2x+y+4z&amp;=3&amp;(1)\\\\\r\n2(5)+(1)+4z&amp;=3&amp;\\\\\r\n11+4z&amp;=3&amp;\\\\\r\n4z&amp;=-8&amp;\\\\\r\nz&amp;=-2 &amp; \\end{array}[\/latex]<\/p>\r\nThe solution is the ordered triple [latex](x,y,z)=(5,1,-2).[\/latex] This solution can be checked and does satisfy all three equations.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSome students prefer to always eliminate [latex]x[\/latex] first so that the approach to each problem is consistent. While this can be a good strategy, it also can cause some of the steps to take more time as we must first multiply some equations by a constant to set up the [latex]x[\/latex] to eliminate.\r\n\r\nIn the following videos, we show more examples of the algebra you may encounter when solving systems with three variables.\r\n\r\nhttps:\/\/youtu.be\/r6htz3gaHZ0\r\n\r\nhttps:\/\/youtu.be\/3RbVSvvRyeI\r\n<h2>Inconsistent and Dependent<\/h2>\r\nThe systems we have seen so far in this section were <strong>consistent and independent.<\/strong>\u00a0A system is consistent if it has at least one solution. Just as with systems of equations in two variables, we may come across an <strong>inconsistent system<\/strong> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The process of elimination will result in a false statement, such as [latex]0=5[\/latex] or some other contradiction. There is also a notion of <strong>dependent system<\/strong>, and it is identified in the same way as with two variables - the elimination process results in a true statement, such as [latex]0=0[\/latex]. This happens when one or more of the equations is \"redundant\" and does not affect the solution set at all. Here is a visual representation of these cases.\r\n<div class=\"textbox shaded\">\r\n<h3>Infinitely Many or No Solutions<\/h3>\r\n<ul>\r\n \t<li>Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex]3=0[\/latex]. Graphically, a system with no solution is represented by three planes with no point in common. This illustration shows a few ways that three planes can have no common intersection point. Note that a solution must be a single point at which all three planes intersect.<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222701\/CNX_Precalc_Figure_09_02_007n2.jpg\" alt=\"Three geometric figures with three planes in each figure. (a) Vertical planes forming a triangle creating 3 lines. (b) 2 parallel planes and one vertical plane cutting them. (c) Parallel planes that don't intersect.\" width=\"487\" height=\"188\" \/>\r\n<ul>\r\n \t<li>Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as [latex]0=0[\/latex]. Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space. The graphic below shows how three planes can intersect to form a line giving the system infinitely many solutions. This system is dependent because if one of the planes was removed, the solution set would still be the same line.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_2386\" align=\"aligncenter\" width=\"224\"]<img class=\"wp-image-2386 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11210154\/Screen-Shot-2016-07-11-at-2.01.28-PM.png\" alt=\"Three vertically aligned planes intersecting at a single line.\" width=\"224\" height=\"217\" \/> Infinitely many solutions.[\/caption]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the following example, we will see how it is possible to have a system with three variables and no solutions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the system of equations.\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{rll}x - 3y+\\phantom{4}z&amp;=\\:4&amp;(1)\\\\ -\\phantom{3}y-4z&amp;=\\:7 &amp;(2)\\\\ \\phantom{11}2y+8z&amp;=\\:-12&amp;(3)\\end{array}\\right.[\/latex]<\/p>\r\n[reveal-answer q=\"982978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982978\"]\r\n\r\nSince [latex]x[\/latex] is already eliminated from equations\u00a0[latex](2)[\/latex] and\u00a0[latex](3),[\/latex] solve them as a two variable system:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl}-y-4z&amp;=\\:7&amp;\\xrightarrow{\\cdot \\; 2} &amp; -2y-8z &amp;=14\\\\\r\n2y+8z&amp;=\\:-12&amp;\\xrightarrow{\\phantom{\\cdot \\; 2}} &amp; 2y+8z&amp;=-12 \\\\\r\n&amp; &amp; &amp; \\text{_______}&amp;\\text{______} \\\\\r\n&amp; &amp; &amp; 0 &amp;=2 \\\\ \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Recall that when we were solving systems with two variables, a contradiction such as [latex]0=2[\/latex] implied that there was no solution to the system. The same is true for three variable systems.<\/p>\r\nThis system has no solution. We can say the system is inconsistent.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe will show another example of using elimination to solve a system in three variables that ends up having no solution in the following video.\r\n\r\nhttps:\/\/youtu.be\/ryNQsWrUoJw\r\n\r\nThe next example shows a dependent system with three variables. In this book we will not describe the solution of dependent systems with three variables other than just to say \"infinite solutions.\" A more complete description is possible, but saved for later math courses.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the system of equations.\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}\\phantom{-10x}-4y+4z=-5&amp;(1)\\\\ -10x + 2y-6z=-6&amp;(2)\\\\ -\\phantom{5}5x-3y+\\phantom{4}z=-8&amp;(3) \\end{array}\\right.[\/latex]<\/p>\r\n[reveal-answer q=\"125373\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"125373\"]\r\n<p style=\"text-align: left;\">Since the first equation is already missing [latex]x,[\/latex] we use equations\u00a0[latex](2)[\/latex] and\u00a0[latex](3),[\/latex] to eliminate\u00a0[latex]x[\/latex] again.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl} -10x+2y-6z&amp;=\\;-6&amp;\\xrightarrow{\\phantom{\\cdot \\; -2}} &amp; -10x+2y-6z &amp;=\\;-6\\\\\r\n-\\phantom{5}5x-3y+\\phantom{4}z&amp;=\\:-8&amp;\\xrightarrow{\\cdot \\; -2} &amp; 10x+6y-2z&amp;=\\;16 \\\\\r\n&amp; &amp; &amp; \\text{_______}&amp;\\text{______} \\\\\r\n&amp; &amp; &amp; 8y-8z&amp;=\\;10 \\ \\ \\ \\ (4) \\\\ \\end{array}[\/latex]<\/p>\r\nWe now solve\u00a0the system consisting of equations [latex](1)[\/latex] and\u00a0[latex](4)[\/latex] by eliminating [latex]y.[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl} -4y+4z&amp;=\\;-5&amp;\\xrightarrow{\\cdot \\; 2} &amp; -8y+8z&amp;=\\;-10\\\\\r\n8y-8z&amp;=\\;10&amp;\\xrightarrow{\\phantom{\\cdot \\; 2}} &amp; 8y-8z&amp;=\\;10 \\\\\r\n&amp; &amp; &amp; \\text{_______}&amp;\\text{______} \\\\\r\n&amp; &amp; &amp; 0&amp;=\\;0 \\\\ \\end{array}[\/latex]<\/p>\r\nSince we reach an identity, the system is dependent and the solution set is infinite. All points along the intersection line will satisfy all three equations. In this book we are not going to describe the solution set in any more detail and will simply say there are \"Infinite Solutions\".\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last video example, we show a system that has an infinite number of solutions. You do not need to watch the part of the video where the infinite solutions are described in more detail using the variable [latex]t,[\/latex] but you can if you are curious.\r\n\r\nhttps:\/\/youtu.be\/mThiwW8nYAU\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>A solution to a system of linear equations in three variables is an ordered triple of the form [latex](x,y,z)[\/latex].<\/li>\r\n \t<li>Solutions can be verified using substitution. A solution needs to satisfy all three equations.<\/li>\r\n \t<li>Systems of three variables can be solved using the same techniques as we used to solve systems with two variables, including elimination and substitution.<\/li>\r\n \t<li>A system with three variables can have one solution, no solution, or infinitely many solutions.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether an ordered triple is a solution of a system of three linear equations with three variables.<\/li>\n<li>Solve systems of three linear equations with three variables.<\/li>\n<li>Determine if a system of three linear equations with three variables is inconsistent or consistent.<\/li>\n<li>Determine if equations are dependent or independent for a system of three linear equations with three variables.<\/li>\n<\/ul>\n<\/div>\n<p>Sometimes in problems or applications there are three (or more) unknowns. A system with three unknowns requires three equations in order to generally have a single solution (although just like with two variables, there are special cases.) We typically use [latex]z[\/latex] for the third variable. An example of a system with three variables is shown here:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\phantom{6}x+\\phantom{4}y+\\phantom{5}z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/p>\n<p>A <strong>solution <\/strong>to a system of three equations in three variables is an ordered triple [latex]\\left(x,y,z\\right)[\/latex] that makes all three equations true. Graphically, the solution set to a single linear equation in three variables can be represented by a plane in three dimensions. A plane is a flat two-dimensional surface that extends infinitely. A system of three equations represents three planes, and a solution represents a common intersection point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2389 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11211012\/Screen-Shot-2016-07-11-at-2.08.59-PM.png\" alt=\"Three planes intersecting at a point.\" width=\"262\" height=\"210\" \/><\/p>\n<p>You can visualize such an intersection by imagining any corner in a rectangular room. A line is defined by the the intersection of two planes: for example, two adjacent walls. A corner is defined by the intersection of three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.<\/p>\n<p>In the first example, we will determine whether an ordered triple is a solution for a system of three linear equations in three variables.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.<\/p>\n<div>[latex]\\begin{array}{l}\\phantom{6}x+\\phantom{4}y+\\phantom{5}z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q642719\">Show Solution<\/span><\/p>\n<div id=\"q642719\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will check each equation by substituting in the values of the ordered triple for [latex]x,y[\/latex], and [latex]z[\/latex].<\/p>\n<p>[latex]\\begin{array}{ccccc}\\begin{array}{r}\\hfill x+y+z=2\\\\ \\hfill \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\hfill \\text{True}\\end{array}& & \\begin{array}{r}\\hfill \\text{}6x - 4y+5z=31\\\\ \\hfill 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ \\hfill 18+8+5=31\\\\ \\hfill \\text{True}\\end{array}& & \\begin{array}{r}\\hfill \\text{}5x+2y+2z=13\\\\ \\hfill 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ \\hfill \\text{}15 - 4+2=13\\\\ \\hfill \\text{True}\\end{array}\\end{array}[\/latex]<\/p>\n<p>The ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system. Note that if any of the three equations was not true, it would not be a solution to the system.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solving a Three-Variable System<\/h2>\n<p>As with many math processes, our goal is to reduce our problem to a problem that was previously solved. If we can eliminate one of the variables, that will result in a system with only two variables and we know how to solve those.<\/p>\n<p>Consider the following example, which begins with [latex]x[\/latex] already removed from two of the equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the system of equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}x-\\phantom{32}y+\\phantom{2}z=5&(1)\\\\\\phantom{x-}-2y+\\phantom{2}z=6&(2)\\\\\\phantom{x-3}2y-\\phantom{2}2z=-12&(3)\\end{array}\\right.[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223787\">Show Solution<\/span><\/p>\n<div id=\"q223787\" class=\"hidden-answer\" style=\"display: none\">\n<p>It is good practice to label the equations as we go because we will need to reference them later. The key observation is that equations [latex](2)[\/latex] and [latex](3)[\/latex] only have the variables [latex]y[\/latex] and [latex]z,[\/latex] so we can solve them as a two variable system and find the solution for\u00a0[latex]y[\/latex] and [latex]z[\/latex] immediately.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{ \\begin{array}{rl}-2y+z&=\\text{ }6\\\\ 2y-2z&=\\text{ }-12\\end{array}\\right.[\/latex]<\/p>\n<p>Adding the equations right away will eliminate [latex]y[\/latex] and allow us to find the value of [latex]z[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rl}\\hfill \\\\ \\:\\:-2y+z&=6\\hfill \\\\ 2y-2z&=-12\\hfill \\end{array}}{\\text{}\\text{}\\text{}\\text{}\\text{}\\:\\:\\:\\:\\:\\:-z=-6}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]z=6[\/latex]<\/p>\n<p>Now we can substitute the value for\u00a0[latex]z[\/latex] that we obtained into equation [latex](2)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl}-2y+(6)&=\\;6\\\\-2y&=\\;6-6\\\\-2y&=\\;0\\\\\\,\\,\\,\\,y&=\\;0\\end{array}[\/latex]<\/p>\n<p>Be careful here not to get confused with a solution of\u00a0[latex]y = 0[\/latex] and an inconsistent solution. It is ok for variables to equal\u00a0[latex]0.[\/latex]<\/p>\n<p>Now we can substitute\u00a0[latex]z = 6[\/latex] and\u00a0[latex]y = 0[\/latex] back into equation [latex](1)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl}x-y+z&=\\;5\\\\x-0+6&=\\;5\\\\x+6&=\\;5\\\\x&=\\;5-6\\\\x&=\\;-1\\end{array}[\/latex]<\/p>\n<p>The solution is the ordered triple [latex](x,y,z)=(-1,0,6).[\/latex] We leave it to you to check that this solution satisfies all three equations.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The strategy we used in the preceding example informs the general process that we will use to solve systems with three unknowns.<\/p>\n<div class=\"textbox shaded\">\n<h3>Solving a linear system with three unknowns<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Pick any pair of equations and eliminate one variable.<\/li>\n<li>Pick another pair of equations and eliminate the same variable.<\/li>\n<li>You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.<\/li>\n<li>Back-substitute known variables into any one of the original equations and solve for the missing variable.<\/li>\n<\/ol>\n<\/div>\n<p>Since our previous example started already in Step 3, we will now do an example where we must do all four of the steps.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve the system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}2x+\\phantom{4}y+4z=3&(1)\\\\ -x + 3y-4z=6&(2)\\\\ 3x-5y+4z=2&(3) \\end{array}\\right.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q535211\">Show Solution<\/span><\/p>\n<div id=\"q535211\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we should identify which variable we plan to eliminate in Steps 1 and 2. Notice that the coefficients for [latex]z[\/latex] are all either [latex]4[\/latex] or\u00a0[latex]-4.[\/latex] This suggests eliminating\u00a0[latex]z[\/latex] in two pairs of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rll}2x+\\phantom{4}y+4z=3&(1) \\\\ -x + 3y-4z=6&(2) \\end{array}}{\\phantom{2}x+4y\\phantom{-24z}=9 \\:\\:\\:\\:(4)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\begin{array}{rll}-x+3y-4z=6&(2) \\\\ 3x -5y+4z=2&(3) \\end{array}}{2x -2y\\phantom{-24z}=8 \\:\\:\\:\\:(5)}[\/latex]<\/p>\n<p style=\"text-align: left;\">Equations [latex](4)[\/latex] and\u00a0[latex](5)[\/latex] form our two variable system,<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{ \\begin{array}{rll}x+4y&=\\:9&(4)\\\\ 2x-2y&=\\:8&(5)\\end{array}\\right.[\/latex]<\/p>\n<p>We now solve this two variable system by elimination.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl}x+4y&=9&\\xrightarrow{\\cdot \\; -2} & -2x-8y &=-18\\\\  2x-2y&=8&\\xrightarrow{\\phantom{\\cdot \\; -2}} & 2x-2y&=8 \\\\  & & & \\text{_______}&\\text{______} \\\\  & & & -10y&=-10 \\\\  & & & y &=1 \\\\ \\end{array}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}x+4y&=9&(4)\\\\  x+4(1)&=9& \\\\  x&=5 & \\end{array}[\/latex]<\/p>\n<p>Now, substitute [latex]x=5, y=1[\/latex] into equation [latex](1).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rll}2x+y+4z&=3&(1)\\\\  2(5)+(1)+4z&=3&\\\\  11+4z&=3&\\\\  4z&=-8&\\\\  z&=-2 & \\end{array}[\/latex]<\/p>\n<p>The solution is the ordered triple [latex](x,y,z)=(5,1,-2).[\/latex] This solution can be checked and does satisfy all three equations.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Some students prefer to always eliminate [latex]x[\/latex] first so that the approach to each problem is consistent. While this can be a good strategy, it also can cause some of the steps to take more time as we must first multiply some equations by a constant to set up the [latex]x[\/latex] to eliminate.<\/p>\n<p>In the following videos, we show more examples of the algebra you may encounter when solving systems with three variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2: System of Three Equations with Three Unknowns Using Elimination\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/r6htz3gaHZ0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1: System of Three Equations with Three Unknowns Using Elimination\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3RbVSvvRyeI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Inconsistent and Dependent<\/h2>\n<p>The systems we have seen so far in this section were <strong>consistent and independent.<\/strong>\u00a0A system is consistent if it has at least one solution. Just as with systems of equations in two variables, we may come across an <strong>inconsistent system<\/strong> of equations in three variables, which means that it does not have a solution that satisfies all three equations. The process of elimination will result in a false statement, such as [latex]0=5[\/latex] or some other contradiction. There is also a notion of <strong>dependent system<\/strong>, and it is identified in the same way as with two variables &#8211; the elimination process results in a true statement, such as [latex]0=0[\/latex]. This happens when one or more of the equations is &#8220;redundant&#8221; and does not affect the solution set at all. Here is a visual representation of these cases.<\/p>\n<div class=\"textbox shaded\">\n<h3>Infinitely Many or No Solutions<\/h3>\n<ul>\n<li>Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex]3=0[\/latex]. Graphically, a system with no solution is represented by three planes with no point in common. This illustration shows a few ways that three planes can have no common intersection point. Note that a solution must be a single point at which all three planes intersect.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01222701\/CNX_Precalc_Figure_09_02_007n2.jpg\" alt=\"Three geometric figures with three planes in each figure. (a) Vertical planes forming a triangle creating 3 lines. (b) 2 parallel planes and one vertical plane cutting them. (c) Parallel planes that don't intersect.\" width=\"487\" height=\"188\" \/><\/p>\n<ul>\n<li>Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as [latex]0=0[\/latex]. Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space. The graphic below shows how three planes can intersect to form a line giving the system infinitely many solutions. This system is dependent because if one of the planes was removed, the solution set would still be the same line.<\/li>\n<\/ul>\n<div id=\"attachment_2386\" style=\"width: 234px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2386\" class=\"wp-image-2386 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11210154\/Screen-Shot-2016-07-11-at-2.01.28-PM.png\" alt=\"Three vertically aligned planes intersecting at a single line.\" width=\"224\" height=\"217\" \/><\/p>\n<p id=\"caption-attachment-2386\" class=\"wp-caption-text\">Infinitely many solutions.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the following example, we will see how it is possible to have a system with three variables and no solutions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{rll}x - 3y+\\phantom{4}z&=\\:4&(1)\\\\ -\\phantom{3}y-4z&=\\:7 &(2)\\\\ \\phantom{11}2y+8z&=\\:-12&(3)\\end{array}\\right.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982978\">Show Solution<\/span><\/p>\n<div id=\"q982978\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]x[\/latex] is already eliminated from equations\u00a0[latex](2)[\/latex] and\u00a0[latex](3),[\/latex] solve them as a two variable system:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl}-y-4z&=\\:7&\\xrightarrow{\\cdot \\; 2} & -2y-8z &=14\\\\  2y+8z&=\\:-12&\\xrightarrow{\\phantom{\\cdot \\; 2}} & 2y+8z&=-12 \\\\  & & & \\text{_______}&\\text{______} \\\\  & & & 0 &=2 \\\\ \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Recall that when we were solving systems with two variables, a contradiction such as [latex]0=2[\/latex] implied that there was no solution to the system. The same is true for three variable systems.<\/p>\n<p>This system has no solution. We can say the system is inconsistent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We will show another example of using elimination to solve a system in three variables that ends up having no solution in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ryNQsWrUoJw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The next example shows a dependent system with three variables. In this book we will not describe the solution of dependent systems with three variables other than just to say &#8220;infinite solutions.&#8221; A more complete description is possible, but saved for later math courses.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\begin{array}{lr}\\phantom{-10x}-4y+4z=-5&(1)\\\\ -10x + 2y-6z=-6&(2)\\\\ -\\phantom{5}5x-3y+\\phantom{4}z=-8&(3) \\end{array}\\right.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125373\">Show Solution<\/span><\/p>\n<div id=\"q125373\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Since the first equation is already missing [latex]x,[\/latex] we use equations\u00a0[latex](2)[\/latex] and\u00a0[latex](3),[\/latex] to eliminate\u00a0[latex]x[\/latex] again.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl} -10x+2y-6z&=\\;-6&\\xrightarrow{\\phantom{\\cdot \\; -2}} & -10x+2y-6z &=\\;-6\\\\  -\\phantom{5}5x-3y+\\phantom{4}z&=\\:-8&\\xrightarrow{\\cdot \\; -2} & 10x+6y-2z&=\\;16 \\\\  & & & \\text{_______}&\\text{______} \\\\  & & & 8y-8z&=\\;10 \\ \\ \\ \\ (4) \\\\ \\end{array}[\/latex]<\/p>\n<p>We now solve\u00a0the system consisting of equations [latex](1)[\/latex] and\u00a0[latex](4)[\/latex] by eliminating [latex]y.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rllrl} -4y+4z&=\\;-5&\\xrightarrow{\\cdot \\; 2} & -8y+8z&=\\;-10\\\\  8y-8z&=\\;10&\\xrightarrow{\\phantom{\\cdot \\; 2}} & 8y-8z&=\\;10 \\\\  & & & \\text{_______}&\\text{______} \\\\  & & & 0&=\\;0 \\\\ \\end{array}[\/latex]<\/p>\n<p>Since we reach an identity, the system is dependent and the solution set is infinite. All points along the intersection line will satisfy all three equations. In this book we are not going to describe the solution set in any more detail and will simply say there are &#8220;Infinite Solutions&#8221;.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last video example, we show a system that has an infinite number of solutions. You do not need to watch the part of the video where the infinite solutions are described in more detail using the variable [latex]t,[\/latex] but you can if you are curious.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/mThiwW8nYAU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>A solution to a system of linear equations in three variables is an ordered triple of the form [latex](x,y,z)[\/latex].<\/li>\n<li>Solutions can be verified using substitution. A solution needs to satisfy all three equations.<\/li>\n<li>Systems of three variables can be solved using the same techniques as we used to solve systems with two variables, including elimination and substitution.<\/li>\n<li>A system with three variables can have one solution, no solution, or infinitely many solutions.<\/li>\n<\/ul>\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-122\">\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>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/Lumen%20Learning\">http:\/\/Lumen%20Learning<\/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\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra: 8.1 Systems of Linear Equations: Gaussian Elimination. <strong>Authored by<\/strong>: Stitz, Carl and Zeager, Jeff. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.stitz-zeager.com\/szca07042013.pdf\">http:\/\/www.stitz-zeager.com\/szca07042013.pdf<\/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: Solve a System of 3 Equations with 3 Unknowns Using Back Substitution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/HHIjTChrIxE\">https:\/\/youtu.be\/HHIjTChrIxE<\/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: System of Three Equations with Three Unknowns Using Elimination. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/r6htz3gaHZ0\">https:\/\/youtu.be\/r6htz3gaHZ0<\/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 1: System of Three Equations with Three Unknowns Using Elimination. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3RbVSvvRyeI\">https:\/\/youtu.be\/3RbVSvvRyeI<\/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\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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