{"id":15361,"date":"2021-10-11T23:05:37","date_gmt":"2021-10-11T23:05:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/summary-solutions-of-systems-of-equations\/"},"modified":"2021-10-16T20:45:25","modified_gmt":"2021-10-16T20:45:25","slug":"summary-solutions-of-systems-of-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/summary-solutions-of-systems-of-equations\/","title":{"raw":"Summary: Solutions to Systems of Equations","rendered":"Summary: Solutions to Systems of Equations"},"content":{"raw":"\n<h2>Key Concepts<\/h2>\n<strong>How to&nbsp;determine whether an ordered pair is a solution to&nbsp;a system of linear equations<\/strong>\n<ol>\n \t<li>Substitute the ordered pair into each equation in the system.<\/li>\n \t<li>Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<strong>Three possible outcomes for solutions to systems of equations<\/strong>\n<ul>\n \t<li><strong>One Solution:<\/strong>&nbsp;When a system of equations intersects at an ordered pair, the system has one solution.<\/li>\n \t<li><strong>Infinite Solutions:<\/strong> Sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions.<\/li>\n \t<li><strong>No Solution:<\/strong> When the lines that make up a system are parallel, there are no solutions because the two lines share no points in common.<\/li>\n<\/ul>\n<strong>Based upon these outcomes, there are three types of systems of linear equations in two variables.<\/strong>\n<div>\n<ul>\n \t<li>An <strong>independent system<\/strong> has exactly <em>one solution<\/em> pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\n \t<li>An <strong>inconsistent system<\/strong> has <em>no solution<\/em>.The two lines are parallel and will never intersect.<\/li>\n \t<li>A <strong>dependent system<\/strong> has<em> infinitely many solutions<\/em>. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<strong>System of linear equations<\/strong>&nbsp; two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.\n\n<strong>The solution&nbsp;<\/strong>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.\n\nA <strong>consistent system<\/strong> of equations has at least one solution.\n\nAn <strong>independent system<\/strong>&nbsp;has a single solution.\n\nA<strong> dependent system<\/strong>&nbsp;has an infinite number of solutions.\n\nAn <strong>inconsistent<\/strong> <strong>system<\/strong>&nbsp;is when there are no points common to both lines.\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1wPTO6N9CIDGjFSlJJfN3dEJo10gtTlt3pDXk2SvSoVc\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>\n","rendered":"<h2>Key Concepts<\/h2>\n<p><strong>How to&nbsp;determine whether an ordered pair is a solution to&nbsp;a system of linear equations<\/strong><\/p>\n<ol>\n<li>Substitute the ordered pair into each equation in the system.<\/li>\n<li>Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<p><strong>Three possible outcomes for solutions to systems of equations<\/strong><\/p>\n<ul>\n<li><strong>One Solution:<\/strong>&nbsp;When a system of equations intersects at an ordered pair, the system has one solution.<\/li>\n<li><strong>Infinite Solutions:<\/strong> Sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions.<\/li>\n<li><strong>No Solution:<\/strong> When the lines that make up a system are parallel, there are no solutions because the two lines share no points in common.<\/li>\n<\/ul>\n<p><strong>Based upon these outcomes, there are three types of systems of linear equations in two variables.<\/strong><\/p>\n<div>\n<ul>\n<li>An <strong>independent system<\/strong> has exactly <em>one solution<\/em> pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\n<li>An <strong>inconsistent system<\/strong> has <em>no solution<\/em>.The two lines are parallel and will never intersect.<\/li>\n<li>A <strong>dependent system<\/strong> has<em> infinitely many solutions<\/em>. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>System of linear equations<\/strong>&nbsp; two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.<\/p>\n<p><strong>The solution&nbsp;<\/strong>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.<\/p>\n<p>A <strong>consistent system<\/strong> of equations has at least one solution.<\/p>\n<p>An <strong>independent system<\/strong>&nbsp;has a single solution.<\/p>\n<p>A<strong> dependent system<\/strong>&nbsp;has an infinite number of solutions.<\/p>\n<p>An <strong>inconsistent<\/strong> <strong>system<\/strong>&nbsp;is when there are no points common to both lines.<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1wPTO6N9CIDGjFSlJJfN3dEJo10gtTlt3pDXk2SvSoVc\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n","protected":false},"author":167848,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"02be1920d6be4d22b2b2141d960c3b59","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15361","chapter","type-chapter","status-publish","hentry"],"part":13184,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15361\/revisions"}],"predecessor-version":[{"id":15550,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15361\/revisions\/15550"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/13184"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15361\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15361"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15361"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15361"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}