{"id":7444,"date":"2014-09-19T16:57:45","date_gmt":"2014-09-19T16:57:45","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=7444"},"modified":"2014-12-02T08:25:25","modified_gmt":"2014-12-02T08:25:25","slug":"math-review-of-solving-systems-of-equations-using-elimination","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2014\/09\/19\/math-review-of-solving-systems-of-equations-using-elimination\/","title":{"rendered":"Math Review of Solving Systems of Equations Using Elimination"},"content":{"rendered":"

Overview<\/h3>\n

One of the methods for solving systems of equations with two variables adds the equations together and then eliminates one variable. It is a useful strategy when solving by substitution would be too cumbersome.<\/p>\n

Adding Systems of Equations<\/h3>\n

Entire systems of equations can be added just as easily as adding polynomials. Suppose one equation in the system is 2r + 3s = 13 and the other equation is 4r – 3s = 17. The 2r and 4r can be added to equal 6r and the 3s and -3s can be added to equal 0. Then 13 and 17 equal 30.<\/p>\n

Figure 1: An example of the process of adding systems of equations.<\/p>\n

<\/a><\/p>\n

Using the Additive Inverse<\/h3>\n

Note that the additive inverse of 3s is -3s, so 3s -3s equals zero. This is a very important key to the elimination method. Adding the equations together when one monomial is the additive inverse of the other eliminates that variable temporarily, so that 6r = 30. If 6r = 30, then r equals 5. If r equals 5, then 2r equals 10, and 10 + 3s = 13, 3s = 13 – 10 or 3 and s equals 1. To check the other equation, 4\u22195 or 20 – 3 = 17.<\/p>\n

Multiply Before Adding<\/h3>\n

In the above example, the additive inverses were clearly stated as 3s and -3s. Sometimes, equations need to be multiplied by a constant to find an equivalent equation. Remember that the graphed lines of equivalent equations, such as x + y = 8, 2x +2 y =16, 3x + 3y = 36 and so on, are still the same line with the same solution set for x and y. Suppose the system of equations were 2a + 3b = 8, a + 3b = 7. The monomials 3b and 3b are not inverses of each other. In order to change 3b to its inverse, multiply every term in the equation by -1 so that the equivalent becomes \u2013a – 3b = -7. 2a- a equals a, 3b – 3b = 0, and 8 – 7 = 1. If a equals 1, then 2 + 3b = 8, and 3b =8-2 =6 so b =2. Similarly, 1 + 6 = 7. Suppose that 3w + 6x = -6 and 5w – 2x = 14. Then 15w -6x = 42, multiplying each monomial term by 3. Adding 3w + 6x = -6 and 15w -6x =42 leaves 18w = 36. If w = 2, then 6 + 6x = -6, or 6x = -6 + -6, 6x = -12, so x = -2. Similarly, 10 + 4 = 14.<\/p>\n

Figure 2: Multiplying by the constant -1 leads to an additive inverse that can eliminate one variable.<\/p>\n

<\/a><\/p>\n

Systems with No Solution<\/h3>\n

Just because systems can be added and then solved using substitution doesn\u2019t mean that all systems of equations have a solution. Suppose the system of equations is x + y = 5 and 2x + 2y = 6. That leads to 2 equations x + y = 5 and x + y = 3, or y = 5 \u2013x and y = 3 – x. If all terms in the equation x + y = 3 were multiplied by -1, then x – x + y \u2013 y = 5 – 3 leading to a false statement 0 = 2. If those linear equations were graphed on a coordinate plane, they would be parallel, showing no common solution.<\/p>\n

Figure 3: Parallel linear equations in the coordinate plane have no common solution.<\/p>\n

<\/a><\/p>\n

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SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Hilton Head Island, SC: visit Tutoring in Hilton Head Island, SC<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Overview One of the methods for solving systems of equations with two variables adds the equations together and then eliminates one variable. It is a useful strategy when solving by substitution would be too cumbersome. Adding Systems of Equations Entire systems of equations can be added just as easily as adding polynomials. Suppose one equation […]<\/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":[2718,2823,2822,2824],"class_list":["post-7444","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-addition-method-of-solving-systems-of-equations","tag-additive-inverse","tag-elimination","tag-multiplying-before-adding"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/7444","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=7444"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/7444\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=7444"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=7444"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=7444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}