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Math 1090 Quiz 3 Solutions February 23rd, 2015 Answer the following question in the space provided. The value of every question is indicated at the beginning. Time: 12 minutes. Name: UID: 1. (20 points) Consider the following system of equations. 3x − 2y − 7z = 0 x−y−z =1 −x + 2y − 3z = a Solve this system for a = −4 and a = 2 and write a one-sentence geometric interpretation of the solution. Solution. The augmented matrix for the system 3 −2 −7 1 −1 −1 −1 2 −3 is 0 1 a Our goal is to apply row operations until the 3 × 3 matrix on the left becomes uppertriangular (namely, until we have zeros below the diagonal. 3 −2 −7 0 1 −1 −1 1 −1 2 −3 a 3 −2 −7 0 [2] ↔ 3[2] − [1] 0 −1 4 3 [3] ↔ [3] + [2] 0 1 −4 a + 1 3 −2 −7 0 0 −1 4 3 [3] ↔ [3] + [2] 0 0 0 a+4 Case a = −4 For a = −4, our augmented matrix becomes 3 −2 −7 0 0 −1 4 3 0 0 0 0 The last equation is 0 = 0, so we are left with only two equations for 3 variables 3x − 2y − 7z = 0 −y + 4z = 3 This means that we will be getting infinitely many solutions. To find them, fix one of the variables, say z, and use the equations o write x and y in terms of z. From the second equation we get y = 4z − 3 and plugging that into the first equation yields y z }| { 3x − 2 (4z − 3) −7z = 0 3x − 8z + 6 − 7z = 0 3x − 15z + 6 = 0 3x = 15z − 6 x = 5z − 2 so our solutions are y x z }| { z }| { (5z − 2, 4z − 3, z) and for every value of z we get one valid solution. For instance, for z = 0, we get the point (−2, −3, 0), and you can check that it satisfies all 3 equations of the original system. Geometric interpretation. Remember that a linear equation in three variables represents a plane. Normally, when you intersect 3 planes you get a point. In this case you are getting infinitely many points, so what this means is that your 3 planes intersect in a line. Case a = 2 For a = 2, our augmented 3 0 0 matrix becomes −2 −7 0 −1 4 3 0 0 6 so the last equation reads 0x + 0y + 0z = 6 0=6 which is impossible. This tells you that you have no solutions and geometrically, that the 3 planes have no points in common. Page 2