{"id":6799,"date":"2014-03-18T15:10:42","date_gmt":"2014-03-18T15:10:42","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=6799"},"modified":"2014-12-02T08:25:31","modified_gmt":"2014-12-02T08:25:31","slug":"math-introduction-to-consistent-and-inconsistent-systems","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/math-introduction-to-consistent-and-inconsistent-systems\/","title":{"rendered":"Math Introduction to Consistent and Inconsistent Systems"},"content":{"rendered":"

Overview:<\/strong><\/p>\n

One of the ways to compare equations in linear\u00a0systems is to see how many solutions both equations have in common. If both equations have no solutions in common, they are referred to as inconsistent. If only one ordered pair can solve both equations, the systems are called consistent, but if the equations have more than one point in common, they are dependent.<\/p>\n

What Does It Mean to Have Solutions in Common?<\/strong><\/p>\n

A system of equations has solutions in common if there is at least one ordered pair that will solve both equations, even if there are many solutions that are not shared.\u00a0 For example, suppose one equation is x + y = 6 and another is x – y = 2.\u00a0 Do they have any solutions in common?\u00a0 There are many solutions to the equation x + y = 6.\u00a0 For example, if x equals 0, y equals 6, if x equals 1, y equals 5, if x equals 2, y equals 4 and so on.\u00a0 Similarly, in the second equation, if x equals 2, y equals 0, if x equals 3, y equals 1, and so on.\u00a0 Both equations do have one solution in common.\u00a0 If x equals 4 and y equals 2 in both equations, both equations have true solutions.<\/p>\n

What Are Inconsistent Systems?<\/strong><\/p>\n

Inconsistent systems of linear equations have no solutions in common.\u00a0 When each equation is graphed on a coordinate plane, the lines are parallel. Suppose one equation is x – y = 8 and another equation is 5x \u2013 5y = 25.\u00a0 The solutions for the first equation are ordered pairs such as {(16, 8), (15, 7), (14, 6), (13, 5), (12, 4) \u2026}.\u00a0 The second equation can be simplified by dividing each member by 5, so that 5x\/5 -5y\/5 = 25\/5.\u00a0 In other words, x – y = 5.\u00a0 They have no solutions in common.<\/p>\n

What Are Consistent Systems?<\/strong><\/p>\n

Consistent systems have at least one solution in common.\u00a0 For example, the equations x + y = 6 and x \u2013 y = 2 have one solution in common, the ordered pair (4, 2) because 4 + 2 equals 6 and 4 – 2 equals 2.\u00a0 Similarly, the equations x + y = 12 and 3y = x have one solution in common, the ordered pair (9, 3), because 9 + 3 = 12 and 3(3) = 9.<\/p>\n

What Are Dependent Systems?<\/strong><\/p>\n

Dependent systems have an infinite number of solutions in common.\u00a0 Both equations can be graphed on the same line.\u00a0 For example, suppose the equations were x – y = 5 and 5x – 5y = 25.\u00a0 The solution set of the first equation would include the points {(5, 0), (6, 1), (7-2), (8,3), \u00a0\u2026}.\u00a0 The solution set of the second equation would also include the points {(5, 0), (6, 1), (7, 2), (8,3),\u2026}.<\/p>\n

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Overview: One of the ways to compare equations in linear\u00a0systems is to see how many solutions both equations have in common. If both equations have no solutions in common, they are referred to as inconsistent. If only one ordered pair can solve both equations, the systems are called consistent, but if the equations have more […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[2636,2638,2637],"class_list":["post-6799","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-consistent-systems","tag-dependent-systems","tag-inconsistent-systems"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/6799","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=6799"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/6799\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=6799"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=6799"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=6799"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}