{"id":1155,"date":"2022-03-19T00:21:34","date_gmt":"2022-03-19T00:21:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1155"},"modified":"2026-01-17T02:26:54","modified_gmt":"2026-01-17T02:26:54","slug":"2-7-1-introduction-to-system-of-linear-equations-in-three-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-7-1-introduction-to-system-of-linear-equations-in-three-variables\/","title":{"raw":"2.7.1: Introduction to Systems of Linear Equations in Three Variables","rendered":"2.7.1: Introduction to Systems of Linear Equations in Three Variables"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-109\" class=\"standard post-109 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Explain the meaning of solving a system of three variable linear equations<\/li>\r\n \t<li>Explain one solution, no solution, and infinitely many solutions graphically with three planes<\/li>\r\n \t<li>Explain one solution, no solution, and infinitely many solutions algebraically when solving a system of three variable linear equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Linear Equations in Three Variables<\/h2>\r\nA linear equation in one variable, which can be written as [latex]ax=b[\/latex], can be graphed as a single point, which has no dimensions (see figure 1). A linear equation in two variables, which can be written as [latex]ax+by=c[\/latex], can be graphed as a line, which has one dimension (i.e., length) (see figure 2). If we add another variable term, we get a linear equation in three variables, which can be written as [latex]ax+by+cz=d[\/latex]. Following the pattern of dimensions, the graph of a linear equation in three variables must have two dimensions, length and width; the graph is a plane (see figure 3). Any solution of a linear equation in three variables is an <em><strong>ordered triple<\/strong><\/em> [latex]\\left(x,y,z\\right)[\/latex]. For example, the equation [latex]2x-y+3z=6[\/latex] has [latex](0, 0, 2),\\;(1, -4, 0)[\/latex], and [latex](1, 2, 2)[\/latex] as three of an infinite number of solutions. Such solutions, and the plane that they make up, are plotted in three-dimensional space, with the [latex]z[\/latex]-axis being perpendicular to both the [latex]x[\/latex]- and [latex]y[\/latex]-axes. We can think of a plane as an infinitely large, two-dimensional, flat sheet of paper.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 17.20154%;\">[latex]ax=b[\/latex]<\/th>\r\n<th style=\"width: 44.287548%;\">[latex]ax+by=c[\/latex]<\/th>\r\n<th style=\"width: 38.382542%;\">\r\n<div class=\"mceTemp\">[latex]ax+by+cz=d[\/latex]<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.20154%;\"><img class=\"alignnone wp-image-1520 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-300x65.png\" alt=\"A point at 1 on a number line.\" width=\"300\" height=\"65\" \/> Figure 1. Solution set of [latex]ax=b[\/latex] is a point.<\/td>\r\n<td style=\"width: 44.287548%;\">[caption id=\"attachment_1208\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1208 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-300x300.png\" alt=\"A line that passes through (0, -3) and (2, 0)\" width=\"300\" height=\"300\" \/> Figure 2. Solution set of [latex]ax+by=c[\/latex]\u00a0 is a line.[\/caption]<\/td>\r\n<td style=\"width: 38.382542%;\">[caption id=\"attachment_1524\" align=\"aligncenter\" width=\"254\"]<img class=\"wp-image-1524 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-254x300.png\" alt=\"A plane in a three-dimensional coordinate space.\" width=\"254\" height=\"300\" \/> Figure 3. Solution set of [latex]ax+by+cz=d[\/latex] is a plane.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine if the ordered triple is a solution of the equation [latex]2x-y+3z=8[\/latex].\r\n\r\n1. (3, \u20132, 0)\r\n\r\n2. (\u20132, 9, 1)\r\n<h4>Solution<\/h4>\r\nTo determine if the ordered triples are solutions, we must substitute the value of each variable into the equation to see if the equation is true.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial; font-size: 1rem;\">1<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial; font-size: 1rem;\">2<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial; font-size: 1rem;\">[latex]\\begin{aligned}2x-y+3z&amp;=8\\\\2(3)-(-2)+3(0)&amp;=8\\\\8&amp;=8\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: initial; font-size: 1rem;\">(3, \u20132, 0) is a solution of the equation.<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial; font-size: 1rem;\">[latex]\\begin{aligned}2x-y+3z&amp;=8\\\\2(-2)-(9)+3(1)&amp;=8\\\\-10&amp;=8\\;\\;\\;\\text{False}\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: initial; font-size: 1rem;\">(\u20132, 9, 1) is not a solution of the equation.<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine if the ordered triple is a solution of the equation [latex]x-4y+2z=6[\/latex].\r\n<ol>\r\n \t<li>(3, \u20132, \u20132)<\/li>\r\n \t<li>(\u20132, 1, 6)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm629\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm629\"]\r\n<ol>\r\n \t<li>(3, \u20132, \u20132) is not a solution.<\/li>\r\n \t<li>(\u20132, 1, 6) is a solution.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Systems of Linear Equations in Three Variables<\/h2>\r\nEach equation in a system of three linear equations in three variables is represented graphically by a plane. The\u00a0<strong><em>solution set<\/em>\u00a0<\/strong>of such\u00a0a system consists of all the points where the three planes meet.\r\n<h3>A single solution<\/h3>\r\nIt is possible for a system to have a single point [latex]\\left(x,y,z\\right)[\/latex] as the solution. Graphically, the ordered triple defines the point that is the intersection of three planes in space. We can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by 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<div class=\"textbox\">\r\n<h3>Solution Set, One Solution<\/h3>\r\nThe figure below\u00a0illustrates how a\u00a0system with three variables\u00a0can have one solution.\u00a0Systems that have a single solution are those which result in a\u00a0<strong>solution set\u00a0<\/strong>consisting of an ordered triple [latex]\\left(x,y,z\\right)[\/latex]. Graphically, the ordered triple defines a point that is the intersection of three planes in space.\r\n<div class=\"wp-nocaption size-full wp-image-2389 aligncenter\">\r\n\r\n[caption id=\"attachment_2389\" align=\"aligncenter\" width=\"262\"]<img class=\"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 crossing each other at right angles and meeting at a single point.\" width=\"262\" height=\"210\" \/> Figure 4. Three planes meeting at a single point.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\nFor an ordered triple to be a solution for a system of three linear equations in three variables, it has to satisfy every equation in the system.\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nDetermine if (1, \u20132, 3) is a solution of the system of equations:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x+y+z&amp;=2\\\\2x+y+z&amp;=3\\\\x-y+2z&amp;=5\\end{aligned}[\/latex]<\/p>\r\n\r\n<h4>Solution<\/h4>\r\nWe need to substitute the given values for each variable and see if the equation is true.\r\n\r\n1st equation:\r\n\r\n[latex]\\begin{aligned}x+y+z&amp;=2\\\\(1)+(-2)+(3)&amp;=2\\\\2&amp;=2\\;\\;\\text{True}\\end{aligned}[\/latex]\r\n\r\n2nd equation:\r\n\r\n[latex]\\begin{aligned}2x+y+z&amp;=3\\\\2(1)+(-2)+(3)&amp;=2\\\\3&amp;=3\\;\\;\\text{True}\\end{aligned}[\/latex]\r\n\r\n3rd equation:\r\n\r\n[latex]\\begin{aligned}x-y+2z&amp;=5\\\\(1)-(-2)+2(3)&amp;=5\\\\9&amp;=5\\;\\;\\text{False}\\end{aligned}[\/latex]\r\n\r\n(1, \u20132, 3) is NOT a solution of the system of equations.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine if (2, \u20131, 4) is a solution of the system of equations:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x+y+z&amp;=5\\\\2x+y+z&amp;=7\\\\x-y+2z&amp;=11\\end{aligned}[\/latex]<\/p>\r\n[reveal-answer q=\"hjm230\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm230\"]\r\n\r\n(2, \u20131, 4) is a solution of the system of equations.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Infinitely Many Solutions<\/h3>\r\nIn a system of two linear equations, we saw that it was possible to have a <em><strong>dependent system<\/strong><\/em>. This occured when the two lines representing the equations coincided resulting in an infinite number of solutions on the line. A similar thing can happen with a system of three linear equations where all three planes coincide and the solution set would be all points on the plane (figure 7). Three planes can also meet in a line (think about three consecutive pages in a book that meet at the spline), which also results in an infinite number of solutions that lie on the line where they meet (figure 5). Another option is when two identical planes intersect a third plane in a line (figure 6). Algebraically, these scenarios show up when we get an identity showing up when we are solving the system.\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-110\" class=\"standard post-110 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>Infinitely Many solutions<\/h3>\r\nSystems 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 (figures 5 and 6) or a plane (figure 7) giving the system infinitely many solutions on the line or on the plane.\r\n<table style=\"border-collapse: collapse; width: 114.045%; height: 230px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 230px;\">\r\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\"><img class=\"wp-image-1950 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02231851\/3-planes-meeting-in-a-line-300x185.png\" alt=\"3 planes meeting in a line\" width=\"304\" height=\"187\" \/> Figure 5. Infinitely many solutions on the line of intersection of three planes.<\/th>\r\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\">&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-1551\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/13233833\/2-coinciding-plane-meeting-plane--300x255.png\" alt=\"Two coincident planes meet a third plane in a line.\" width=\"249\" height=\"212\" \/><\/th>\r\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\"><img class=\"aligncenter wp-image-1552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/13233836\/3-coinciding-planes-300x271.png\" alt=\"Three coincident planes.\" width=\"250\" height=\"226\" \/> Figure 7. Three coincident planes. Infinite solutions on the plane.<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h3>No Solution<\/h3>\r\nJust as with systems of equations in two variables, we may come across an\u00a0<strong><em>inconsistent system<\/em>\u00a0<\/strong>of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a contradiction when we try to solve the system algebraically.\r\n<div class=\"textbox shaded\">\r\n<h3>NO SOLUTION<\/h3>\r\nSystems 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. Three parallel planes (figure 8),\u00a0two parallel planes and one intersecting plane (figure 9), two coincident planes and a parallel plane (figure 10), and three planes that intersect the other two but not at the same location (figure 11).\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">All Planes Are Parallel<\/th>\r\n<th style=\"width: 50%; text-align: center;\">Not All Planes Are Parallel<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1953\" align=\"aligncenter\" width=\"251\"]<img class=\"wp-image-1953\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234617\/3-parallel-planes-300x243.png\" alt=\"Three parallel planes.\" width=\"251\" height=\"203\" \/> Figure 8. Three parallel lines[\/caption]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1954\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1954\" style=\"background-color: #ffffff;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234621\/2-c-planes-and-1-parallel-300x243.png\" alt=\"Two coincident planes parallel to another plane.\" width=\"250\" height=\"202\" \/> Figure 10. Two coincident planes parallel to another plane.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1952\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1952\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234612\/no-soln-planes-300x244.png\" alt=\"Two parallel planes meeting another plane in two lines.\" width=\"250\" height=\"204\" \/> Figure 9. Two parallel planes meeting another plane.[\/caption]\r\n\r\n[caption id=\"attachment_1956\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/03001002\/3-planes-no-soln-300x254.png\" alt=\"Three independent planes intersecting each other in three lines.\" width=\"250\" height=\"212\" \/> Figure 11. Three independent planes with no points in common.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"wp-nocaption aligncenter\"><\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nA system of 3 linear equations that is being solved algebraically ends up with an equation that is an identity. Describe the two graphical scenarios that would cause this.\r\n<h4>Solution<\/h4>\r\nAn identity occurs when the system is dependent. This means that either the three planes coincide, 2 of the three planes coincide, or the three planes meet in a line.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nWhat can be said about a system of three linear equations in three variables when two of the equations in the system are equivalent? What does this mean graphically?\r\n\r\n[reveal-answer q=\"hjm162\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm162\"]\r\n\r\nThe system is dependent and has infinite solutions.\r\n\r\nTwo planes are identical so the third plane must cross the identical planes in a line.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-109\" class=\"standard post-109 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Explain the meaning of solving a system of three variable linear equations<\/li>\n<li>Explain one solution, no solution, and infinitely many solutions graphically with three planes<\/li>\n<li>Explain one solution, no solution, and infinitely many solutions algebraically when solving a system of three variable linear equations<\/li>\n<\/ul>\n<\/div>\n<h2>Linear Equations in Three Variables<\/h2>\n<p>A linear equation in one variable, which can be written as [latex]ax=b[\/latex], can be graphed as a single point, which has no dimensions (see figure 1). A linear equation in two variables, which can be written as [latex]ax+by=c[\/latex], can be graphed as a line, which has one dimension (i.e., length) (see figure 2). If we add another variable term, we get a linear equation in three variables, which can be written as [latex]ax+by+cz=d[\/latex]. Following the pattern of dimensions, the graph of a linear equation in three variables must have two dimensions, length and width; the graph is a plane (see figure 3). Any solution of a linear equation in three variables is an <em><strong>ordered triple<\/strong><\/em> [latex]\\left(x,y,z\\right)[\/latex]. For example, the equation [latex]2x-y+3z=6[\/latex] has [latex](0, 0, 2),\\;(1, -4, 0)[\/latex], and [latex](1, 2, 2)[\/latex] as three of an infinite number of solutions. Such solutions, and the plane that they make up, are plotted in three-dimensional space, with the [latex]z[\/latex]-axis being perpendicular to both the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-axes. We can think of a plane as an infinitely large, two-dimensional, flat sheet of paper.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 17.20154%;\">[latex]ax=b[\/latex]<\/th>\n<th style=\"width: 44.287548%;\">[latex]ax+by=c[\/latex]<\/th>\n<th style=\"width: 38.382542%;\">\n<div class=\"mceTemp\">[latex]ax+by+cz=d[\/latex]<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 17.20154%;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1520 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-300x65.png\" alt=\"A point at 1 on a number line.\" width=\"300\" height=\"65\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-300x65.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-65x14.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-225x49.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D-350x76.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/0-D.png 642w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/> Figure 1. Solution set of [latex]ax=b[\/latex] is a point.<\/td>\n<td style=\"width: 44.287548%;\">\n<div id=\"attachment_1208\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1208\" class=\"wp-image-1208 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-300x300.png\" alt=\"A line that passes through (0, -3) and (2, 0)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y61.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1208\" class=\"wp-caption-text\">Figure 2. Solution set of [latex]ax+by=c[\/latex]\u00a0 is a line.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 38.382542%;\">\n<div id=\"attachment_1524\" style=\"width: 264px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1524\" class=\"wp-image-1524 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-254x300.png\" alt=\"A plane in a three-dimensional coordinate space.\" width=\"254\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-254x300.png 254w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-768x906.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-868x1024.png 868w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-65x77.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-225x266.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane-350x413.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/plane.png 1144w\" sizes=\"auto, (max-width: 254px) 100vw, 254px\" \/><\/p>\n<p id=\"caption-attachment-1524\" class=\"wp-caption-text\">Figure 3. Solution set of [latex]ax+by+cz=d[\/latex] is a plane.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine if the ordered triple is a solution of the equation [latex]2x-y+3z=8[\/latex].<\/p>\n<p>1. (3, \u20132, 0)<\/p>\n<p>2. (\u20132, 9, 1)<\/p>\n<h4>Solution<\/h4>\n<p>To determine if the ordered triples are solutions, we must substitute the value of each variable into the equation to see if the equation is true.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial; font-size: 1rem;\">1<\/p>\n<\/td>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial; font-size: 1rem;\">2<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial; font-size: 1rem;\">[latex]\\begin{aligned}2x-y+3z&=8\\\\2(3)-(-2)+3(0)&=8\\\\8&=8\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: initial; font-size: 1rem;\">(3, \u20132, 0) is a solution of the equation.<\/p>\n<\/td>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial; font-size: 1rem;\">[latex]\\begin{aligned}2x-y+3z&=8\\\\2(-2)-(9)+3(1)&=8\\\\-10&=8\\;\\;\\;\\text{False}\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: initial; font-size: 1rem;\">(\u20132, 9, 1) is not a solution of the equation.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine if the ordered triple is a solution of the equation [latex]x-4y+2z=6[\/latex].<\/p>\n<ol>\n<li>(3, \u20132, \u20132)<\/li>\n<li>(\u20132, 1, 6)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm629\">Show Answer<\/span><\/p>\n<div id=\"qhjm629\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(3, \u20132, \u20132) is not a solution.<\/li>\n<li>(\u20132, 1, 6) is a solution.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Systems of Linear Equations in Three Variables<\/h2>\n<p>Each equation in a system of three linear equations in three variables is represented graphically by a plane. The\u00a0<strong><em>solution set<\/em>\u00a0<\/strong>of such\u00a0a system consists of all the points where the three planes meet.<\/p>\n<h3>A single solution<\/h3>\n<p>It is possible for a system to have a single point [latex]\\left(x,y,z\\right)[\/latex] as the solution. Graphically, the ordered triple defines the point that is the intersection of three planes in space. We can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by 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<div class=\"textbox\">\n<h3>Solution Set, One Solution<\/h3>\n<p>The figure below\u00a0illustrates how a\u00a0system with three variables\u00a0can have one solution.\u00a0Systems that have a single solution are those which result in a\u00a0<strong>solution set\u00a0<\/strong>consisting of an ordered triple [latex]\\left(x,y,z\\right)[\/latex]. Graphically, the ordered triple defines a point that is the intersection of three planes in space.<\/p>\n<div class=\"wp-nocaption size-full wp-image-2389 aligncenter\">\n<div id=\"attachment_2389\" style=\"width: 272px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2389\" class=\"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 crossing each other at right angles and meeting at a single point.\" width=\"262\" height=\"210\" \/><\/p>\n<p id=\"caption-attachment-2389\" class=\"wp-caption-text\">Figure 4. Three planes meeting at a single point.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>For an ordered triple to be a solution for a system of three linear equations in three variables, it has to satisfy every equation in the system.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Determine if (1, \u20132, 3) is a solution of the system of equations:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x+y+z&=2\\\\2x+y+z&=3\\\\x-y+2z&=5\\end{aligned}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>We need to substitute the given values for each variable and see if the equation is true.<\/p>\n<p>1st equation:<\/p>\n<p>[latex]\\begin{aligned}x+y+z&=2\\\\(1)+(-2)+(3)&=2\\\\2&=2\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\n<p>2nd equation:<\/p>\n<p>[latex]\\begin{aligned}2x+y+z&=3\\\\2(1)+(-2)+(3)&=2\\\\3&=3\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\n<p>3rd equation:<\/p>\n<p>[latex]\\begin{aligned}x-y+2z&=5\\\\(1)-(-2)+2(3)&=5\\\\9&=5\\;\\;\\text{False}\\end{aligned}[\/latex]<\/p>\n<p>(1, \u20132, 3) is NOT a solution of the system of equations.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine if (2, \u20131, 4) is a solution of the system of equations:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x+y+z&=5\\\\2x+y+z&=7\\\\x-y+2z&=11\\end{aligned}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm230\">Show Answer<\/span><\/p>\n<div id=\"qhjm230\" class=\"hidden-answer\" style=\"display: none\">\n<p>(2, \u20131, 4) is a solution of the system of equations.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Infinitely Many Solutions<\/h3>\n<p>In a system of two linear equations, we saw that it was possible to have a <em><strong>dependent system<\/strong><\/em>. This occured when the two lines representing the equations coincided resulting in an infinite number of solutions on the line. A similar thing can happen with a system of three linear equations where all three planes coincide and the solution set would be all points on the plane (figure 7). Three planes can also meet in a line (think about three consecutive pages in a book that meet at the spline), which also results in an infinite number of solutions that lie on the line where they meet (figure 5). Another option is when two identical planes intersect a third plane in a line (figure 6). Algebraically, these scenarios show up when we get an identity showing up when we are solving the system.<\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-110\" class=\"standard post-110 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>Infinitely Many solutions<\/h3>\n<p>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 (figures 5 and 6) or a plane (figure 7) giving the system infinitely many solutions on the line or on the plane.<\/p>\n<table style=\"border-collapse: collapse; width: 114.045%; height: 230px;\">\n<tbody>\n<tr style=\"height: 230px;\">\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1950 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02231851\/3-planes-meeting-in-a-line-300x185.png\" alt=\"3 planes meeting in a line\" width=\"304\" height=\"187\" \/> Figure 5. Infinitely many solutions on the line of intersection of three planes.<\/th>\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\">&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1551\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/13233833\/2-coinciding-plane-meeting-plane--300x255.png\" alt=\"Two coincident planes meet a third plane in a line.\" width=\"249\" height=\"212\" \/><\/th>\n<th style=\"width: 33.3333%; vertical-align: bottom; height: 230px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/13233836\/3-coinciding-planes-300x271.png\" alt=\"Three coincident planes.\" width=\"250\" height=\"226\" \/> Figure 7. Three coincident planes. Infinite solutions on the plane.<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3>No Solution<\/h3>\n<p>Just as with systems of equations in two variables, we may come across an\u00a0<strong><em>inconsistent system<\/em>\u00a0<\/strong>of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a contradiction when we try to solve the system algebraically.<\/p>\n<div class=\"textbox shaded\">\n<h3>NO SOLUTION<\/h3>\n<p>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. Three parallel planes (figure 8),\u00a0two parallel planes and one intersecting plane (figure 9), two coincident planes and a parallel plane (figure 10), and three planes that intersect the other two but not at the same location (figure 11).<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">All Planes Are Parallel<\/th>\n<th style=\"width: 50%; text-align: center;\">Not All Planes Are Parallel<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1953\" style=\"width: 261px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1953\" class=\"wp-image-1953\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234617\/3-parallel-planes-300x243.png\" alt=\"Three parallel planes.\" width=\"251\" height=\"203\" \/><\/p>\n<p id=\"caption-attachment-1953\" class=\"wp-caption-text\">Figure 8. Three parallel lines<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_1954\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1954\" class=\"wp-image-1954\" style=\"background-color: #ffffff;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234621\/2-c-planes-and-1-parallel-300x243.png\" alt=\"Two coincident planes parallel to another plane.\" width=\"250\" height=\"202\" \/><\/p>\n<p id=\"caption-attachment-1954\" class=\"wp-caption-text\">Figure 10. Two coincident planes parallel to another plane.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1952\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1952\" class=\"wp-image-1952\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02234612\/no-soln-planes-300x244.png\" alt=\"Two parallel planes meeting another plane in two lines.\" width=\"250\" height=\"204\" \/><\/p>\n<p id=\"caption-attachment-1952\" class=\"wp-caption-text\">Figure 9. Two parallel planes meeting another plane.<\/p>\n<\/div>\n<div id=\"attachment_1956\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1956\" class=\"wp-image-1956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/03001002\/3-planes-no-soln-300x254.png\" alt=\"Three independent planes intersecting each other in three lines.\" width=\"250\" height=\"212\" \/><\/p>\n<p id=\"caption-attachment-1956\" class=\"wp-caption-text\">Figure 11. Three independent planes with no points in common.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"wp-nocaption aligncenter\"><\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>A system of 3 linear equations that is being solved algebraically ends up with an equation that is an identity. Describe the two graphical scenarios that would cause this.<\/p>\n<h4>Solution<\/h4>\n<p>An identity occurs when the system is dependent. This means that either the three planes coincide, 2 of the three planes coincide, or the three planes meet in a line.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>What can be said about a system of three linear equations in three variables when two of the equations in the system are equivalent? What does this mean graphically?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm162\">Show Answer<\/span><\/p>\n<div id=\"qhjm162\" class=\"hidden-answer\" style=\"display: none\">\n<p>The system is dependent and has infinite solutions.<\/p>\n<p>Two planes are identical so the third plane must cross the identical planes in a line.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/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-1155\">\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>Authored by<\/strong>: Hazel McKenna. <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><li>Learning Outcomes. <strong>Authored by<\/strong>: Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Linear Equations in Three Variables. FIgures 1, 2, 3, 5, 6, 7, 8, 9 10, 11. Try It: hjm162; hjm230; hjm629.. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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":422608,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Hazel McKenna\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Learning Outcomes\",\"author\":\"Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Linear Equations in Three Variables. FIgures 1, 2, 3, 5, 6, 7, 8, 9 10, 11. 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