{"id":388,"date":"2019-07-15T22:44:40","date_gmt":"2019-07-15T22:44:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/inconsistent-and-dependent-systems-in-three-variables\/"},"modified":"2019-07-15T22:44:40","modified_gmt":"2019-07-15T22:44:40","slug":"inconsistent-and-dependent-systems-in-three-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/inconsistent-and-dependent-systems-in-three-variables\/","title":{"raw":"Classify Solutions to Systems in Three Variables","rendered":"Classify Solutions to Systems in Three Variables"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Recognize whether a system has one, none, or an infinite number of solutions based on its solution.<\/li>\n \t<li>Use correct notation to express solutions to systems of three equations.<\/li>\n<\/ul>\n<\/div>\nJust 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 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 false statement, such as [latex]3=7[\/latex] or some other contradiction.\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nAs you discovered when solving systems that have one solution, well-organized work is essential to being certain about the result you obtain. It can take several steps for the contradiction in a system with no solution to appear, as in the example below. Patient effort is essential.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inconsistent System of Three Equations in Three Variables<\/h3>\nSolve the following system.\n<p style=\"text-align: center\">[latex]\\begin{align}x - 3y+z=4 &amp;&amp; \\left(1\\right) \\\\ -x+2y - 5z=3 &amp;&amp; \\left(2\\right) \\\\ 5x - 13y+13z=8 &amp;&amp; \\left(3\\right) \\end{align}[\/latex]<\/p>\n[reveal-answer q=\"721134\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"721134\"]\n\nLooking at the coefficients of [latex]x[\/latex], we can see that we can eliminate [latex]x[\/latex] by adding equation (1) to equation (2).\n<p style=\"text-align: center\">[latex]\\begin{align}x - 3y+z=4 \\\\ -x+2y - 5z=3 \\\\ \\hline -y - 4z=7\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align} (1) \\\\ (2) \\\\ (4) \\end{align}[\/latex]<\/p>\nNext, we multiply equation (1) by [latex]-5[\/latex] and add it to equation (3).\n<p style=\"text-align: center\">[latex]\\begin{align}\u22125x+15y\u22125z&amp;=\u221220 \\\\ 5x\u221213y+13z&amp;=8 \\\\ \\hline 2y+8z&amp;=\u221212\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&amp;(1)\\text{ multiplied by }\u22125 \\\\ &amp;(3) \\\\ &amp;(5) \\end{align}[\/latex]<\/p>\nThen, we multiply equation (4) by 2 and add it to equation (5).\n<p style=\"text-align: center\">[latex]\\begin{align}\u22122y\u22128z&amp;=14 \\\\ 2y+8z&amp;=\u221212 \\\\ \\hline 0&amp;=2\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&amp;(4)\\text{ multiplied by }2 \\\\ &amp;(5) \\\\&amp; \\end{align}[\/latex]<\/p>\nThe final equation [latex]0=2[\/latex] is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.\n<h4>Analysis of the Solution<\/h4>\nIn this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.\n\n[\/hidden-answer]\n\n<\/div>\nWatch the video below for another example of using elimination in a system that has no solution.\nhttps:\/\/youtu.be\/ryNQsWrUoJw\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSolve the system of three equations in three variables.\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ \\text{ }y - 3z=1\\hfill \\\\ 2x+y+5z=0\\hfill \\end{array}[\/latex]<\/p>\n[reveal-answer q=\"563392\"]Show Solution[\/reveal-answer]\n\n[hidden-answer a=\"563392\"]\n\nNo solution.\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Expressing the Solution of a System of Dependent Equations Containing Three Variables<\/h2>\nWe know from working with systems of equations in two variables that a <strong>dependent system<\/strong> of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nAs you did with dependent two-by-two systems, you can write the general solution to a three-by-three system. See the example below for a demonstration, the try the following problem. It will take practice before it feels natural to write the general solution to a dependent system.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Solution to a Dependent System of Equations<\/h3>\nFind the solution to the given system of three equations in three variables.\n<p style=\"text-align: center\">[latex]\\begin{align}2x+y - 3z=0 &amp;&amp; \\left(1\\right)\\\\ 4x+2y - 6z=0 &amp;&amp; \\left(2\\right)\\\\ x-y+z=0 &amp;&amp; \\left(3\\right)\\end{align}[\/latex]<\/p>\n[reveal-answer q=\"633686\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"633686\"]\n\nFirst, we can multiply equation (1) by [latex]-2[\/latex] and add it to equation (2).\n<p style=\"text-align: center\">[latex]\\begin{align} \u22124x\u22122y+6z=0 &amp;\\hspace{9mm} (1)\\text{ multiplied by }\u22122 \\\\ 4x+2y\u22126z=0 &amp;\\hspace{9mm} (2) \\end{align}[\/latex]<\/p>\nWe do not need to proceed any further. The result we get is an identity, [latex]0=0[\/latex], which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by [latex]-2[\/latex], and adding it to equation (1). We then perform the same steps as above and find the same result, [latex]0=0[\/latex].\n\nWhen a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have\n<p style=\"text-align: center\">[latex]\\begin{align}2x+y\u22123z=0 \\\\ x\u2212y+z=0 \\\\ \\hline 3x\u22122z=0 \\end{align}[\/latex]<\/p>\nWe then solve the resulting equation for [latex]z[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{align}3x - 2z=0 \\\\ z=\\frac{3}{2}x \\end{align}[\/latex]<\/p>\nWe back-substitute the expression for [latex]z[\/latex] into one of the equations and solve for [latex]y[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;2x+y - 3\\left(\\frac{3}{2}x\\right)=0 \\\\ &amp;2x+y-\\frac{9}{2}x=0 \\\\ &amp;y=\\frac{9}{2}x - 2x \\\\ &amp;y=\\frac{5}{2}x \\end{align}[\/latex]<\/p>\nSo the general solution is [latex]\\left(x,\\frac{5}{2}x,\\frac{3}{2}x\\right)[\/latex]. In this solution, [latex]x[\/latex] can be any real number. The values of [latex]y[\/latex] and [latex]z[\/latex] are dependent on the value selected for [latex]x[\/latex].\n<h4>Analysis of the Solution<\/h4>\nAs shown below, two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03185121\/CNX_Precalc_Figure_09_02_0092.jpg\" alt=\"Two overlapping planes intersecting a third. The first overlapping plane's equation is negative 4x minus 2y plus 6z equals zero. The second overlapping plane's equation is 4x plus 2y minus 6z equals zero. The third plane's equation is x minus y plus z equals zero.\" width=\"487\" height=\"288\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Does the generic solution to a dependent system always have to be written in terms of [latex]x?[\/latex]<\/h4>\n<em>No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of [latex]x[\/latex] and if needed [latex]x[\/latex] and [latex]y[\/latex].<\/em>\n\n<\/div>\nSee the following video for another example of a dependent three-by-three system.\nhttps:\/\/youtu.be\/mThiwW8nYAU\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSolve the following system.\n<p style=\"text-align: center\">[latex]\\begin{gathered}x+y+z=7 \\\\ 3x - 2y-z=4 \\\\ x+6y+5z=24 \\end{gathered}[\/latex]<\/p>\n[reveal-answer q=\"195958\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"195958\"]\n\nInfinitely many number of solutions of the form [latex]\\left(x,4x - 11,-5x+18\\right)[\/latex].\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29695&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Recognize whether a system has one, none, or an infinite number of solutions based on its solution.<\/li>\n<li>Use correct notation to express solutions to systems of three equations.<\/li>\n<\/ul>\n<\/div>\n<p>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 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 false statement, such as [latex]3=7[\/latex] or some other contradiction.<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>As you discovered when solving systems that have one solution, well-organized work is essential to being certain about the result you obtain. It can take several steps for the contradiction in a system with no solution to appear, as in the example below. Patient effort is essential.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inconsistent System of Three Equations in Three Variables<\/h3>\n<p>Solve the following system.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}x - 3y+z=4 && \\left(1\\right) \\\\ -x+2y - 5z=3 && \\left(2\\right) \\\\ 5x - 13y+13z=8 && \\left(3\\right) \\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q721134\">Show Solution<\/span><\/p>\n<div id=\"q721134\" class=\"hidden-answer\" style=\"display: none\">\n<p>Looking at the coefficients of [latex]x[\/latex], we can see that we can eliminate [latex]x[\/latex] by adding equation (1) to equation (2).<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}x - 3y+z=4 \\\\ -x+2y - 5z=3 \\\\ \\hline -y - 4z=7\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align} (1) \\\\ (2) \\\\ (4) \\end{align}[\/latex]<\/p>\n<p>Next, we multiply equation (1) by [latex]-5[\/latex] and add it to equation (3).<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\u22125x+15y\u22125z&=\u221220 \\\\ 5x\u221213y+13z&=8 \\\\ \\hline 2y+8z&=\u221212\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&(1)\\text{ multiplied by }\u22125 \\\\ &(3) \\\\ &(5) \\end{align}[\/latex]<\/p>\n<p>Then, we multiply equation (4) by 2 and add it to equation (5).<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\u22122y\u22128z&=14 \\\\ 2y+8z&=\u221212 \\\\ \\hline 0&=2\\end{align}[\/latex][latex]\\hspace{5mm} \\begin{align}&(4)\\text{ multiplied by }2 \\\\ &(5) \\\\& \\end{align}[\/latex]<\/p>\n<p>The final equation [latex]0=2[\/latex] is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>In this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the video below for another example of using elimination in a system that has no solution.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the system of three equations in three variables.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{ }x+y+z=2\\hfill \\\\ \\text{ }y - 3z=1\\hfill \\\\ 2x+y+5z=0\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q563392\">Show Solution<\/span><\/p>\n<div id=\"q563392\" class=\"hidden-answer\" style=\"display: none\">\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Expressing the Solution of a System of Dependent Equations Containing Three Variables<\/h2>\n<p>We know from working with systems of equations in two variables that a <strong>dependent system<\/strong> of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.<\/p>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>As you did with dependent two-by-two systems, you can write the general solution to a three-by-three system. See the example below for a demonstration, the try the following problem. It will take practice before it feels natural to write the general solution to a dependent system.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Solution to a Dependent System of Equations<\/h3>\n<p>Find the solution to the given system of three equations in three variables.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}2x+y - 3z=0 && \\left(1\\right)\\\\ 4x+2y - 6z=0 && \\left(2\\right)\\\\ x-y+z=0 && \\left(3\\right)\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q633686\">Show Solution<\/span><\/p>\n<div id=\"q633686\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we can multiply equation (1) by [latex]-2[\/latex] and add it to equation (2).<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} \u22124x\u22122y+6z=0 &\\hspace{9mm} (1)\\text{ multiplied by }\u22122 \\\\ 4x+2y\u22126z=0 &\\hspace{9mm} (2) \\end{align}[\/latex]<\/p>\n<p>We do not need to proceed any further. The result we get is an identity, [latex]0=0[\/latex], which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by [latex]-2[\/latex], and adding it to equation (1). We then perform the same steps as above and find the same result, [latex]0=0[\/latex].<\/p>\n<p>When a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}2x+y\u22123z=0 \\\\ x\u2212y+z=0 \\\\ \\hline 3x\u22122z=0 \\end{align}[\/latex]<\/p>\n<p>We then solve the resulting equation for [latex]z[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}3x - 2z=0 \\\\ z=\\frac{3}{2}x \\end{align}[\/latex]<\/p>\n<p>We back-substitute the expression for [latex]z[\/latex] into one of the equations and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&2x+y - 3\\left(\\frac{3}{2}x\\right)=0 \\\\ &2x+y-\\frac{9}{2}x=0 \\\\ &y=\\frac{9}{2}x - 2x \\\\ &y=\\frac{5}{2}x \\end{align}[\/latex]<\/p>\n<p>So the general solution is [latex]\\left(x,\\frac{5}{2}x,\\frac{3}{2}x\\right)[\/latex]. In this solution, [latex]x[\/latex] can be any real number. The values of [latex]y[\/latex] and [latex]z[\/latex] are dependent on the value selected for [latex]x[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>As shown below, two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03185121\/CNX_Precalc_Figure_09_02_0092.jpg\" alt=\"Two overlapping planes intersecting a third. The first overlapping plane's equation is negative 4x minus 2y plus 6z equals zero. The second overlapping plane's equation is 4x plus 2y minus 6z equals zero. The third plane's equation is x minus y plus z equals zero.\" width=\"487\" height=\"288\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Does the generic solution to a dependent system always have to be written in terms of [latex]x?[\/latex]<\/h4>\n<p><em>No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of [latex]x[\/latex] and if needed [latex]x[\/latex] and [latex]y[\/latex].<\/em><\/p>\n<\/div>\n<p>See the following video for another example of a dependent three-by-three system.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the following system.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}x+y+z=7 \\\\ 3x - 2y-z=4 \\\\ x+6y+5z=24 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q195958\">Show Solution<\/span><\/p>\n<div id=\"q195958\" class=\"hidden-answer\" style=\"display: none\">\n<p>Infinitely many number of solutions of the form [latex]\\left(x,4x - 11,-5x+18\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm29695\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29695&#38;theme=oea&#38;iframe_resize_id=ohm29695&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-388\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 29695. <strong>Authored by<\/strong>: Shahbazian, Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution).. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ryNQsWrUoJw\">https:\/\/youtu.be\/ryNQsWrUoJw<\/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 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions).. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mThiwW8nYAU\">https:\/\/youtu.be\/mThiwW8nYAU<\/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":17533,"menu_order":20,"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\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 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