{"id":5867,"date":"2013-05-07T05:34:09","date_gmt":"2013-05-07T05:34:09","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=5867"},"modified":"2014-12-02T08:27:02","modified_gmt":"2014-12-02T08:27:02","slug":"linear-systems-in-algebra","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/linear-systems-in-algebra\/","title":{"rendered":"Linear Systems in Algebra"},"content":{"rendered":"

Overview:\u00a0 Solving Linear Systems<\/strong>
\nIn order to solve systems of two sentences, the solution has to be true for both sentences.\u00a0 For example, the solution of a problem such as 2x-3y=13 and 3x +y = 3 has to work fox x and y all throughout the problem.\u00a0 Linear systems can be solved by graphing, by substitution, and by addition.<\/p>\n

Solving by Graphing<\/strong>
\nA number sentence that is linear can be graphed along a straight line that contains all the ordered pairs (x, y) that are contained in its solution set.\u00a0 The second sentence in the system (in this example, 3x +y = 3) can also be graphed along its straight line.\u00a0 The point at which both lines intersect is the solution for both sentences.<\/p>\n

Solving by Substitution<\/strong>
\nHowever, if the student has neither graph paper handy or a graphing calculator to find the exact point where lines intersect, there is another way to solve the problem, by substituting one variable in the first sentence with the definition in the second sentence.<\/p>\n

For example, 2x-3y = 13, 3x +y = 3.\u00a0 We already know from the second sentence that y=3 -3x\u00a0 by subtracting 3x from each side of the equation, as in 3x-3x +y = 3 – 3x.<\/p>\n

By process of substitution, 2x -3(3 – 3x) = 13 , or 2x – 9 -9x =13, or 2x – 9 + 9x = 13 + 9 (because a negative of a negative equals a positive), or 11x = 22.\u00a0 Therefore, x = 2.\u00a0 Using the second equation, 6 + y = 3, or y = 6 – 3, or y= -3.<\/p>\n

Checking, 4 + 9 (as a negative times a negative is a positive) =13.<\/p>\n

Solving by Addition<\/strong><\/p>\n

In this example, solve the pair x – 2y = 7, x + y =-2.\u00a0 Adding them together from left to right x + x – 2y + y = 7-2 , or 2x -y = 5.\u00a0 If x equals 1, then y equals -3, because 1 +6 equals 7, making the first pair true. In the second pair, 1-3 equals -2, which makes that also true.\u00a0 Checking the addition sentence,\u00a0 2+3 (as the double negative changes to a positive) equals 5.<\/p>\n

Consistent, Inconsistent, Dependent Systems<\/strong>
\nThe definition of consistent, inconsistent, and dependent systems refers back to the coordinate plane.\u00a0 If two equations have only one point in common, then they are consistent.\u00a0 For example, \u00a02x – 3y = 13 and 3x +y =3, are consistent as they only have one point in common, (2, -3).\u00a0 Similarly, the equations x – 2y =7 and x + y =-2 are consistent, as they have one point in common (1, -3).\u00a0 However, the equations x +y =3 and x +y =-4 have no points in common, and there is no solution set in real space that will solve both equations, because there are no combinations of x and y that will simultaneously solve both equations.\u00a0 They are inconsistent.\u00a0 Dependent systems have an infinite number of points in common, such as x +y =3\u00a0 and 2x +2y = 6,<\/p>\n

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Overview:\u00a0 Solving Linear Systems In order to solve systems of two sentences, the solution has to be true for both sentences.\u00a0 For example, the solution of a problem such as 2x-3y=13 and 3x +y = 3 has to work fox x and y all throughout the problem.\u00a0 Linear systems can be solved by graphing, by […]<\/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":[389,476,923,1047,1752],"class_list":["post-5867","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-consistent","tag-dependent","tag-inconsistent","tag-linear-systems","tag-substitution"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5867","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=5867"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5867\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=5867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=5867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=5867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}