{"id":6811,"date":"2014-03-21T17:57:55","date_gmt":"2014-03-21T17:57:55","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=6811"},"modified":"2014-12-02T08:25:31","modified_gmt":"2014-12-02T08:25:31","slug":"math-introduction-to-solving-problems-with-three-variables","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2014\/03\/21\/math-introduction-to-solving-problems-with-three-variables\/","title":{"rendered":"Math Introduction to Solving Problems with Three Variables"},"content":{"rendered":"

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

Solving systems of equations with three variables is similar to solving systems of equations with two variables.\u00a0 The difference is that substitution occurs in two steps rather than in one step.<\/p>\n

Are There Any Variables That Can Be Eliminated?<\/strong><\/p>\n

In order to solve a system in three variables, first check to see if one variable can be eliminated.\u00a0 If there is a solution for one variable, it can sometimes be solved by direct substitution.\u00a0 Suppose the system of equations were x = 5, y – z = 3 and x + y + z = 18.\u00a0 The variable x can be replaced in the third equation to result in y + z = 13 and y – z = 3. \u00a0Subtracting those two equations [(y +z)-(y-z)] = 13 \u2013 3, results in \u00a0\u00a02z = 10 or z = 5.\u00a0 If y – z equals 3, and z equals 5, then 5 + 3 equals 8.\u00a0 Therefore, x equals 5, y equals 8, and z equals 5.<\/p>\n

Are There Any Equations That Can Be Solved?<\/strong><\/p>\n

Sometimes equivalent equations can be substituted to eliminate a variable.\u00a0 Suppose the equations in the system are x – 2y + 3z = -3; 2x – 3y – z = 7; and 3x + y -2z = 6.\u00a0 The variable y can be eliminated, leaving the equations 7x – z = 9 and 11 – 7z = 25.\u00a0 Then the variable z can be eliminated from the system by multiplying -7(7x-z) = 9(-7) or -49x + 7z = -63; leaving -38x = -38, or x equal to 1.\u00a0 If 7x – z = 9, and 7x is equal to 7, then z is equal to -2, because 7 \u2013 (-2) = 9.\u00a0 If 1 – 2y – 6 = -3, then -2y = 2, and y equals -1.\u00a0 The values can be substituted in all the equations to check the answers.<\/p>\n

How Can Three Variables Be Represented?<\/strong><\/p>\n

If two variables x and y can be represented on a single plane, three variables can be represented in three different planes.\u00a0 Suppose there is a single solution for all three equations.\u00a0 The intersection of all three planes will converge at a single point.\u00a0 If there are many solutions, the planes may coincide, or converge in a line.\u00a0 If the variables have no solutions in common, the planes are parallel to one another.<\/p>\n

What Is an Ordered Triple?<\/strong><\/p>\n

An ordered triple is a set of coordinates that contains the solution for each variable. If the problems can be solved, it may be a single point, or may contain many solutions.\u00a0 Sometimes it is easier to visualize what the solution set might look like. The ordered triple for the equations where x equals 5, y equals 8, and z equals 5 is (5, 8, 5) at a single point.<\/p>\n

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Overview: Solving systems of equations with three variables is similar to solving systems of equations with two variables.\u00a0 The difference is that substitution occurs in two steps rather than in one step. Are There Any Variables That Can Be Eliminated? In order to solve a system in three variables, first check to see if one […]<\/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":[2641,2642,1753],"class_list":["post-6811","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-equations-in-three-variables","tag-ordered-triples","tag-substitution-method"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6811","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=6811"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6811\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=6811"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=6811"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=6811"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}