{"id":704,"date":"2025-10-09T15:33:08","date_gmt":"2025-10-09T15:33:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=704"},"modified":"2026-05-26T14:15:25","modified_gmt":"2026-05-26T14:15:25","slug":"triple-integration-and-applications-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integration-and-applications-background-youll-need-2\/","title":{"raw":"Triple Integration and Applications: Background You'll Need 2","rendered":"Triple Integration and Applications: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Solve three equations with three different variables<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Systems of Three Equations in Three Variables<\/h2>\r\nA system of three equations in three variables helps us solve problems that involve three unknown quantities. Each equation in the system takes the form [latex]ax + by + cz = d[\/latex], where [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] are the variables we're solving for, and [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are numbers that can't all be zero. When we graph any single equation from this system, it creates a plane in three-dimensional space.\r\n<div class=\"page\" title=\"Page 829\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\"><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>systems of three equations in three variables<\/h3>\r\nA system of linear equations with three variables is a set of linear equations with three variables.\r\n\r\n<\/section>In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination.\r\n\r\nThe goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution [latex](x,y,z)[\/latex], which we call an <strong>ordered triple<\/strong>. A system in upper triangular form looks like the following:\r\n\r\n<center>[latex]\\begin{align}Ax+By+Cz &amp;=D \\\\ Ey+Fz &amp;=G \\\\ Hz&amp;=K\\end{align}[\/latex]<\/center>The third equation can be solved for [latex]z[\/latex], and then we back-substitute to find [latex]y[\/latex] and [latex]x[\/latex].\r\n<h3>Number of Solutions<\/h3>\r\nWhen we solve a system of linear equations with three variables, we have many possible solutions.\r\n\r\nGraphically, if there is one solution, the ordered triple defines the point that is the intersection of three planes in space. You 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<ul>\r\n \t<li>Systems that have a single solution is written as an ordered triple [latex]\\left\\{\\left(x,y,z\\right)\\right\\}[\/latex].<\/li>\r\n \t<li aria-label=\"Recall\">Systems that have an infinite number of solutions are those which, after elimination, result in an identity expression that is always true, such as [latex]0=0[\/latex].<\/li>\r\n \t<li aria-label=\"Recall\">Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex]3=0[\/latex].<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_2349\" align=\"aligncenter\" width=\"1258\"]<img class=\"wp-image-2349 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/24213912\/Screenshot-2024-07-24-at-2.39.05%E2%80%AFPM.png\" alt=\"\" width=\"1258\" height=\"1430\" \/> Diagram of a system of linear equations with three variables and its possible solutions[\/caption]\r\n\r\nThe solutions are summarized in the table below.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">In the context of systems of equations, the terms \"consistent,\" \"inconsistent,\" and \"dependent\" are used to describe the nature of the solutions:\r\n<ol>\r\n \t<li><strong>Consistent System<\/strong>: A system of equations is consistent if it has at least one solution. This can be either a single unique solution or infinitely many solutions.<\/li>\r\n \t<li><strong>Inconsistent System<\/strong>: A system of equations is inconsistent if it has no solutions. This occurs when the equations represent parallel lines that never intersect.<\/li>\r\n \t<li><strong>Dependent System<\/strong>: A system of equations is dependent if it has infinitely many solutions. This occurs when the equations represent the same line (they are essentially the same equation written in different forms).<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n<section aria-label=\"Try It\">\r\n<h2 aria-label=\"Try It\">Solve Systems of Three Equations in Three Variables<\/h2>\r\nSolving a system with three variables is very similar to solving one with two variables. It is important to keep track of your work as the addition of one more equation creates more steps in the solution process.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a linear system of three equations, solve for three unknowns.\r\n<\/strong>\r\n<ol>\r\n \t<li>Pick any pair of equations and solve for one variable.<\/li>\r\n \t<li>Pick another pair of equations and solve for 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<\/section><section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"954288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"954288\"]\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<p style=\"text-align: center;\">[latex]\\begin{align} x+y+z=2\\\\ \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\text{True}\\end{align}\\hspace{5mm}[\/latex] [latex]\\hspace{5mm}\\begin{align} 6x - 4y+5z=31\\\\ 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ 18+8+5=31\\\\ \\text{True}\\end{align}\\hspace{5mm}[\/latex] [latex]\\hspace{5mm}\\begin{align}5x+2y+2z=13\\\\ 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ 15 - 4+2=13\\\\ \\text{True}\\end{align}[\/latex]<\/p>\r\nThe ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]325823[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the given system.<center>[latex]\\displaystyle\\begin{cases}x-\\dfrac{1}{3}y+\\dfrac{1}{2}z=1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y-\\dfrac{1}{2}z=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=-1\\end{cases}[\/latex]<\/center>\r\n[reveal-answer q=\"538379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"538379\"]The third equation states that\u00a0[latex]z = \u22121[\/latex], so\u00a0we substitute this into the second equation to obtain a solution for\u00a0[latex]y[\/latex].\r\n<center>[latex]\\begin{array}{l}y-\\dfrac{1}{2}(-1)=4\\\\y+\\dfrac{1}{2}=4\\\\y=4-\\dfrac{1}{2}\\\\y=\\dfrac{8}{2}-\\dfrac{1}{2}\\\\y=\\dfrac{7}{2}\\end{array}[\/latex]<\/center>\r\nNow we have two of our solutions, and we can substitute them both into the first equation to solve for\u00a0[latex]x[\/latex].\r\n<center>[latex]\\begin{array}{l}x-\\dfrac{1}{3}\\left(\\dfrac{7}{2}\\right)+\\dfrac{1}{2}\\left(-1\\right)=1\\\\x-\\dfrac{7}{6}-\\dfrac{1}{2}=1\\\\x-\\dfrac{7}{6}-\\dfrac{3}{6}=1\\\\x-\\dfrac{10}{6}=1\\\\x=1+\\dfrac{10}{6}\\\\x=\\dfrac{6}{6}+\\dfrac{10}{6}\\\\x=\\dfrac{16}{6}=\\dfrac{8}{3}\\end{array}[\/latex]<\/center>Now we have our ordered triple; remember to place each variable solution in order.<center>[latex](x,y,z)=\\left(\\dfrac{8}{3},\\dfrac{7}{2},-1\\right)[\/latex]<\/center><strong>Analysis of the Solution<\/strong>Each of the lines in the system above represents a plane (think about a sheet of paper). If you imagine three sheets of notebook paper each representing a portion of these planes, you will start to see the complexities involved in how three such planes can intersect. Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line, so our intersection point is where all three of these lines meet.\r\n\r\n[caption id=\"attachment_2377\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11200619\/Screen-Shot-2016-07-11-at-1.04.41-PM-300x237.png\" alt=\"Three Planes Intersecting.\" width=\"300\" height=\"237\" \/> Three planes intersecting.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]325825[\/ohm_question]<\/section><\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Solve three equations with three different variables<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Systems of Three Equations in Three Variables<\/h2>\n<p>A system of three equations in three variables helps us solve problems that involve three unknown quantities. Each equation in the system takes the form [latex]ax + by + cz = d[\/latex], where [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] are the variables we&#8217;re solving for, and [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are numbers that can&#8217;t all be zero. When we graph any single equation from this system, it creates a plane in three-dimensional space.<\/p>\n<div class=\"page\" title=\"Page 829\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>systems of three equations in three variables<\/h3>\n<p>A system of linear equations with three variables is a set of linear equations with three variables.<\/p>\n<\/section>\n<p>In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination.<\/p>\n<p>The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution [latex](x,y,z)[\/latex], which we call an <strong>ordered triple<\/strong>. A system in upper triangular form looks like the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}Ax+By+Cz &=D \\\\ Ey+Fz &=G \\\\ Hz&=K\\end{align}[\/latex]<\/div>\n<p>The third equation can be solved for [latex]z[\/latex], and then we back-substitute to find [latex]y[\/latex] and [latex]x[\/latex].<\/p>\n<h3>Number of Solutions<\/h3>\n<p>When we solve a system of linear equations with three variables, we have many possible solutions.<\/p>\n<p>Graphically, if there is one solution, the ordered triple defines the point that is the intersection of three planes in space. You 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<ul>\n<li>Systems that have a single solution is written as an ordered triple [latex]\\left\\{\\left(x,y,z\\right)\\right\\}[\/latex].<\/li>\n<li aria-label=\"Recall\">Systems that have an infinite number of solutions are those which, after elimination, result in an identity expression that is always true, such as [latex]0=0[\/latex].<\/li>\n<li aria-label=\"Recall\">Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as [latex]3=0[\/latex].<\/li>\n<\/ul>\n<figure id=\"attachment_2349\" aria-describedby=\"caption-attachment-2349\" style=\"width: 1258px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2349 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/24213912\/Screenshot-2024-07-24-at-2.39.05%E2%80%AFPM.png\" alt=\"\" width=\"1258\" height=\"1430\" \/><figcaption id=\"caption-attachment-2349\" class=\"wp-caption-text\">Diagram of a system of linear equations with three variables and its possible solutions<\/figcaption><\/figure>\n<p>The solutions are summarized in the table below.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">In the context of systems of equations, the terms &#8220;consistent,&#8221; &#8220;inconsistent,&#8221; and &#8220;dependent&#8221; are used to describe the nature of the solutions:<\/p>\n<ol>\n<li><strong>Consistent System<\/strong>: A system of equations is consistent if it has at least one solution. This can be either a single unique solution or infinitely many solutions.<\/li>\n<li><strong>Inconsistent System<\/strong>: A system of equations is inconsistent if it has no solutions. This occurs when the equations represent parallel lines that never intersect.<\/li>\n<li><strong>Dependent System<\/strong>: A system of equations is dependent if it has infinitely many solutions. This occurs when the equations represent the same line (they are essentially the same equation written in different forms).<\/li>\n<\/ol>\n<\/section>\n<\/div>\n<section aria-label=\"Try It\">\n<h2 aria-label=\"Try It\">Solve Systems of Three Equations in Three Variables<\/h2>\n<p>Solving a system with three variables is very similar to solving one with two variables. It is important to keep track of your work as the addition of one more equation creates more steps in the solution process.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a linear system of three equations, solve for three unknowns.<br \/>\n<\/strong><\/p>\n<ol>\n<li>Pick any pair of equations and solve for one variable.<\/li>\n<li>Pick another pair of equations and solve for 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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Determine whether the ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is a solution to the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ 6x - 4y+5z=31\\hfill \\\\ 5x+2y+2z=13\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q954288\">Show Solution<\/button><\/p>\n<div id=\"q954288\" 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 style=\"text-align: center;\">[latex]\\begin{align} x+y+z=2\\\\ \\left(3\\right)+\\left(-2\\right)+\\left(1\\right)=2\\\\ \\text{True}\\end{align}\\hspace{5mm}[\/latex] [latex]\\hspace{5mm}\\begin{align} 6x - 4y+5z=31\\\\ 6\\left(3\\right)-4\\left(-2\\right)+5\\left(1\\right)=31\\\\ 18+8+5=31\\\\ \\text{True}\\end{align}\\hspace{5mm}[\/latex] [latex]\\hspace{5mm}\\begin{align}5x+2y+2z=13\\\\ 5\\left(3\\right)+2\\left(-2\\right)+2\\left(1\\right)=13\\\\ 15 - 4+2=13\\\\ \\text{True}\\end{align}[\/latex]<\/p>\n<p>The ordered triple [latex]\\left(3,-2,1\\right)[\/latex] is indeed a solution to the system.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm325823\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=325823&theme=lumen&iframe_resize_id=ohm325823&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the given system.<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\begin{cases}x-\\dfrac{1}{3}y+\\dfrac{1}{2}z=1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y-\\dfrac{1}{2}z=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z=-1\\end{cases}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q538379\">Show Solution<\/button><\/p>\n<div id=\"q538379\" class=\"hidden-answer\" style=\"display: none\">The third equation states that\u00a0[latex]z = \u22121[\/latex], so\u00a0we substitute this into the second equation to obtain a solution for\u00a0[latex]y[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-\\dfrac{1}{2}(-1)=4\\\\y+\\dfrac{1}{2}=4\\\\y=4-\\dfrac{1}{2}\\\\y=\\dfrac{8}{2}-\\dfrac{1}{2}\\\\y=\\dfrac{7}{2}\\end{array}[\/latex]<\/div>\n<p>Now we have two of our solutions, and we can substitute them both into the first equation to solve for\u00a0[latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x-\\dfrac{1}{3}\\left(\\dfrac{7}{2}\\right)+\\dfrac{1}{2}\\left(-1\\right)=1\\\\x-\\dfrac{7}{6}-\\dfrac{1}{2}=1\\\\x-\\dfrac{7}{6}-\\dfrac{3}{6}=1\\\\x-\\dfrac{10}{6}=1\\\\x=1+\\dfrac{10}{6}\\\\x=\\dfrac{6}{6}+\\dfrac{10}{6}\\\\x=\\dfrac{16}{6}=\\dfrac{8}{3}\\end{array}[\/latex]<\/div>\n<p>Now we have our ordered triple; remember to place each variable solution in order.<\/p>\n<div style=\"text-align: center;\">[latex](x,y,z)=\\left(\\dfrac{8}{3},\\dfrac{7}{2},-1\\right)[\/latex]<\/div>\n<p><strong>Analysis of the Solution<\/strong>Each of the lines in the system above represents a plane (think about a sheet of paper). If you imagine three sheets of notebook paper each representing a portion of these planes, you will start to see the complexities involved in how three such planes can intersect. Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line, so our intersection point is where all three of these lines meet.<\/p>\n<figure id=\"attachment_2377\" aria-describedby=\"caption-attachment-2377\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11200619\/Screen-Shot-2016-07-11-at-1.04.41-PM-300x237.png\" alt=\"Three Planes Intersecting.\" width=\"300\" height=\"237\" \/><figcaption id=\"caption-attachment-2377\" class=\"wp-caption-text\">Three planes intersecting.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm325825\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=325825&theme=lumen&iframe_resize_id=ohm325825&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-704","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/704","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/704\/revisions"}],"predecessor-version":[{"id":2877,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/704\/revisions\/2877"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/434"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/704\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=704"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=704"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=704"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}