{"id":9083,"date":"2017-01-22T20:39:51","date_gmt":"2017-01-22T20:39:51","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=9083"},"modified":"2017-01-22T20:39:51","modified_gmt":"2017-01-22T20:39:51","slug":"math-review-of-solving-systems-by-addition","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2017\/01\/22\/math-review-of-solving-systems-by-addition\/","title":{"rendered":"Math Review of Solving Systems by Addition"},"content":{"rendered":"

Overview<\/strong><\/h4>\n

Besides solving systems of equations by graphing and substitution, systems of equations can also be solved by addition.\u00a0 Math students can choose the best method for the problem at hand. \u00a0Sometimes this process is called “solving systems of equations by elimination”.<\/p>\n

Using Addition<\/strong><\/h4>\n

Although some systems of equations can be solved by substitution, other systems can be solved by adding both equations. Both equations must be written in standard form as Ax +By =C.\u00a0 For example, suppose the equations in the system are x +y = 5 and 2x \u2013y = 4.\u00a0 They can be expressed as x +2x +y \u2013y = 5 +4.\u00a0 Now x +2x equals 3x, y-y = 0, and 5 +4 = 9.\u00a0 The new equation is 3x +0 = 9, or x =3.\u00a0 If 3 +y = 5, then y equals 2.\u00a0 Checking the solution in the second equation, 2x or 6 -2 does equal 4.\u00a0 The addition method can be used because the addition property for equations states that we can add the same number to both sides of the equation, and the equations are still equivalent expressions, and make a true statement.<\/p>\n

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Using Multiplication, Then Addition<\/strong><\/h4>\n

The multiplication property of equations is an extension of the addition property, since multiplication is repeated addition.\u00a0 Therefore, we can multiply each side of the equation by the same number or expression.\u00a0 This is especially useful to eliminate a variable, before using the addition method.\u00a0 For example, suppose the equations in a system were 5x +3y = 17 and 5x-2y =-3.\u00a0 If they were to be added without multiplying, the new equation would be 5x +5x +3y \u2013 2y = 17-3, or 10x-y =14.\u00a0 That is not closer to a solution, as there are still 2 variables in the system.\u00a0 Suppose both sides of the second equation are multiplied by -1, so that the new equation is -1(5x -2y) = -1(-3), or -5x +2y = 3.\u00a0 Using the addition method, 5x \u2013 5x +3y +2y = 17 +3, or 5y =20, or y =4.\u00a0 If 5x +12 = 17, then 5x = 5, or x =1.\u00a0 Using the second equation to check, 5-8 = -3.<\/p>\n

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Using Multiplication More than Once<\/strong><\/h4>\n

Sometimes the multiplication property needs to be used more than once in order to use the addition method.\u00a0 Suppose the system of equations is 5x +3y = 2 and 3x +5y =-2.\u00a0 Using the multiplication property once, 5(5x +3y) = 5(2) = 25x +15y = 10.\u00a0 The second equation can be multiplied by -3, so that\u00a0\u00a0 \u00a0-3(3x +5y) = -3(-2), or -9x -15y =6.\u00a0 Then the addition method can be used, so that 25x -9x +15y -15y = 10+6 or 16x = 16, or x =1.\u00a0 Solving for y, 3y = 2-5, or -3, so y= -1.\u00a0 Checking the second equation, 3 (1) + 5 (-1) = -2.\u00a0 The solution is (1, -1).<\/p>\n

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Problem-Solving Using the Addition Method<\/strong><\/h4>\n

Suppose that the problem were to translate a word problem to a system of equations, then solve. \u00a0The sum of two numbers is 115, and their difference is 21.\u00a0 Understanding the problem, the first equation is x +y = 115, and the second equation is x-y = 21.\u00a0 Using the addition method, x + x +y-y = 115 +21, or 2x =136, or x = 68.\u00a0 Substituting for x, 68 +y = 115, or y = 47.\u00a0 Checking with the second equation, 68-47 = 21.<\/p>\n

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 <\/p>\n","protected":false},"excerpt":{"rendered":"

Overview Besides solving systems of equations by graphing and substitution, systems of equations can also be solved by addition.\u00a0 Math students can choose the best method for the problem at hand. \u00a0Sometimes this process is called “solving systems of equations by elimination”. Using Addition Although some systems of equations can be solved by substitution, other […]<\/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":[2718,3991,3992],"class_list":["post-9083","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-addition-method-of-solving-systems-of-equations","tag-solving-systems-of-equations-by-elimination","tag-using-multiplication-to-solve-systems-of-equations"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/9083","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=9083"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/9083\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=9083"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=9083"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=9083"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}