{"id":18169,"date":"2020-04-21T04:28:33","date_gmt":"2020-04-21T04:28:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/?post_type=chapter&p=18169"},"modified":"2024-05-01T18:59:14","modified_gmt":"2024-05-01T18:59:14","slug":"summary-solutions-of-systems-of-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/summary-solutions-of-systems-of-equations\/","title":{"raw":"Summary: Solutions to Systems of Equations","rendered":"Summary: Solutions to Systems of Equations"},"content":{"raw":"

Key Concepts<\/h2>\r\nHow to\u00a0determine whether an ordered pair is a solution to\u00a0a system of linear equations<\/strong>\r\n
    \r\n \t
  1. Substitute the ordered pair into each equation in the system.<\/li>\r\n \t
  2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\r\n<\/ol>\r\nThree possible outcomes for solutions to systems of equations<\/strong>\r\n
      \r\n \t
    • One Solution:<\/strong>\u00a0When a system of equations intersects at an ordered pair, the system has one solution.<\/li>\r\n \t
    • Infinite Solutions:<\/strong> Sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions.<\/li>\r\n \t
    • 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>\r\n<\/ul>\r\nBased upon these outcomes, there are three types of systems of linear equations in two variables.<\/strong>\r\n
      \r\n
        \r\n \t
      • An independent system<\/strong> has exactly one solution<\/em> pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\r\n \t
      • An inconsistent system<\/strong> has no solution<\/em>.The two lines are parallel and will never intersect.<\/li>\r\n \t
      • A dependent system<\/strong> has 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>\r\n<\/ul>\r\n<\/div>\r\n

        Glossary<\/h2>\r\nSystem of linear equations<\/strong>\u00a0 two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.\r\n\r\nThe solution\u00a0<\/strong>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.\r\n\r\nA consistent system<\/strong> of equations has at least one solution.\r\n\r\nAn independent system<\/strong>\u00a0has a single solution.\r\n\r\nA dependent system<\/strong>\u00a0has an infinite number of solutions.\r\n\r\nAn inconsistent<\/strong> system<\/strong>\u00a0is when there are no points common to both lines.","rendered":"

        Key Concepts<\/h2>\n

        How to\u00a0determine whether an ordered pair is a solution to\u00a0a system of linear equations<\/strong><\/p>\n

          \n
        1. Substitute the ordered pair into each equation in the system.<\/li>\n
        2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n

          Three possible outcomes for solutions to systems of equations<\/strong><\/p>\n

            \n
          • One Solution:<\/strong>\u00a0When a system of equations intersects at an ordered pair, the system has one solution.<\/li>\n
          • 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
          • 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

            Based upon these outcomes, there are three types of systems of linear equations in two variables.<\/strong><\/p>\n

            \n
              \n
            • An independent system<\/strong> has exactly one solution<\/em> pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\n
            • An inconsistent system<\/strong> has no solution<\/em>.The two lines are parallel and will never intersect.<\/li>\n
            • A dependent system<\/strong> has 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

              Glossary<\/h2>\n

              System of linear equations<\/strong>\u00a0 two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously.<\/p>\n

              The solution\u00a0<\/strong>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.<\/p>\n

              A consistent system<\/strong> of equations has at least one solution.<\/p>\n

              An independent system<\/strong>\u00a0has a single solution.<\/p>\n

              A dependent system<\/strong>\u00a0has an infinite number of solutions.<\/p>\n

              An inconsistent<\/strong> system<\/strong>\u00a0is when there are no points common to both lines.<\/p>\n","protected":false},"author":253111,"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-18169","chapter","type-chapter","status-publish","hentry"],"part":16192,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/18169","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/253111"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/18169\/revisions"}],"predecessor-version":[{"id":19058,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/18169\/revisions\/19058"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16192"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/18169\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=18169"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=18169"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=18169"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=18169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}