{"id":139,"date":"2025-02-13T22:44:14","date_gmt":"2025-02-13T22:44:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule\/"},"modified":"2026-03-24T19:06:28","modified_gmt":"2026-03-24T19:06:28","slug":"solving-systems-with-cramers-rule","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule\/","title":{"raw":"Solving Systems with Cramer's Rule: Learn It 1","rendered":"Solving Systems with Cramer&#8217;s Rule: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate 2 \u00d7 2 and 3 \u00d7 3 determinants.<\/li>\r\n \t<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\r\n \t<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nWe have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.\r\n<h2>Evaluating the Determinant of a 2\u00d72 Matrix<\/h2>\r\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>determinant of a 2 x 2 matrix<\/h3>\r\nThe <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\r\nis defined as\r\n\r\n<img class=\"aligncenter wp-image-5027 \" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1-300x88.png\" alt=\"A diagram showing that the determinant of a 2-by-2 matrix is found by multiplying the top-left and bottom-right entries, then subtracting the product of the top-right and bottom-left entries.\" width=\"235\" height=\"69\" \/>\r\n\r\nNotice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"149250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149250\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&amp;=\\left\\rvert\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right\\rvert\\hfill \\\\ &amp;=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &amp;=27\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321764[\/ohm_question]<\/section><section aria-label=\"Try It\">\r\n<h2>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h2>\r\nFinding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\r\n\r\nFind the <strong>determinant<\/strong> of the 3\u00d73 matrix.\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right][\/latex]<\/div>\r\n<ol>\r\n \t<li>Augment [latex]A[\/latex] with the first two columns.\r\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div><\/li>\r\n \t<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\r\n \t<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\r\n<\/ol>\r\n<img class=\"wp-image-5030 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-300x119.png\" alt=\"A diagram illustrating the diagonal method for finding the determinant of a 3-by-3 matrix. It shows the three downward diagonals whose products are added, and the three upward diagonals whose products are subtracted.\" width=\"398\" height=\"158\" \/>\r\n\r\nThe algebra is as follows:\r\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the 3 \u00d7 3 matrix given\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"915069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915069\"]\r\n\r\nAugment the matrix with the first two columns and then follow the formula. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&amp;=\\left\\rvert\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 &amp; 2\\\\ 3 &amp; -1\\\\ 4 &amp; 0\\end{array}\\right\\rvert\\hfill \\\\ &amp;=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &amp;=0+8+0+4 - 0-6\\hfill \\\\ &amp;=6\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]315407[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321765[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Can we use the same method to find the determinant of a larger matrix?\r\n<\/strong>Yes, but for larger matrices it is best to use a graphing utility or computer software.<\/section><\/section><\/section>\r\n<dl id=\"fs-id1674068\" class=\"definition\">\r\n \t<dd id=\"fs-id1674074\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate 2 \u00d7 2 and 3 \u00d7 3 determinants.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.<\/p>\n<h2>Evaluating the Determinant of a 2\u00d72 Matrix<\/h2>\n<p>A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>determinant of a 2 x 2 matrix<\/h3>\n<p>The <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p>is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5027\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1-300x88.png\" alt=\"A diagram showing that the determinant of a 2-by-2 matrix is found by multiplying the top-left and bottom-right entries, then subtracting the product of the top-right and bottom-left entries.\" width=\"235\" height=\"69\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1-300x88.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1-65x19.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1-225x66.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184246\/10.4.L.1.Diagram1.png 325w\" sizes=\"(max-width: 235px) 100vw, 235px\" \/><\/p>\n<p>Notice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q149250\">Show Solution<\/button><\/p>\n<div id=\"q149250\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&=\\left\\rvert\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right\\rvert\\hfill \\\\ &=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &=27\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321764\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321764&theme=lumen&iframe_resize_id=ohm321764&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<h2>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h2>\n<p>Finding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.<\/p>\n<p>Find the <strong>determinant<\/strong> of the 3\u00d73 matrix.<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right][\/latex]<\/div>\n<ol>\n<li>Augment [latex]A[\/latex] with the first two columns.\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div>\n<\/li>\n<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\n<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5030 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-300x119.png\" alt=\"A diagram illustrating the diagonal method for finding the determinant of a 3-by-3 matrix. It shows the three downward diagonals whose products are added, and the three upward diagonals whose products are subtracted.\" width=\"398\" height=\"158\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-300x119.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-65x26.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-225x90.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2-350x139.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03184421\/10.4.L.1.Diagram2.png 462w\" sizes=\"(max-width: 398px) 100vw, 398px\" \/><\/p>\n<p>The algebra is as follows:<\/p>\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the determinant of the 3 \u00d7 3 matrix given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q915069\">Show Solution<\/button><\/p>\n<div id=\"q915069\" class=\"hidden-answer\" style=\"display: none\">\n<p>Augment the matrix with the first two columns and then follow the formula. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&=\\left\\rvert\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 & 2\\\\ 3 & -1\\\\ 4 & 0\\end{array}\\right\\rvert\\hfill \\\\ &=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &=0+8+0+4 - 0-6\\hfill \\\\ &=6\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm315407\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=315407&theme=lumen&iframe_resize_id=ohm315407&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321765\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321765&theme=lumen&iframe_resize_id=ohm321765&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Can we use the same method to find the determinant of a larger matrix?<br \/>\n<\/strong>Yes, but for larger matrices it is best to use a graphing utility or computer software.<\/section>\n<\/section>\n<\/section>\n<dl id=\"fs-id1674068\" class=\"definition\">\n<dd id=\"fs-id1674074\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":21,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"learn_it","content_attributions":[{"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":""}],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/139"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/139\/revisions"}],"predecessor-version":[{"id":5998,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/139\/revisions\/5998"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/514"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/139\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=139"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=139"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=139"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}