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**Unformatted text preview: **Math 136 Assignment 3 Solutions 1. For each of the following systems of linear equations: i) Write the augmented matrix. ii) Row-reduce the augmented matrix into row echelon form. iii) Find the general solution of the system or explain why the system is inconsistent. a) x 1 + x 2 =- 7 2 x 1 + 4 x 2 + x 3 =- 16 x 1 + 2 x 2 + x 3 = 9 Solution: i) 1 1 0- 7 2 4 1- 16 1 2 1 9 . ii) 1 1 0- 7 2 4 1- 16 1 2 1 9 r 2- 2 r 1 r 3- r 1 1 1 0- 7 0 2 1- 2 0 1 1 16 r 2 r 3 1 1 0- 7 0 1 1 16 0 2 1- 2 r 3- 2 r 2 1 1- 7 0 1 1 16 0 0- 1- 34 (- 1) r 3 1 1 0- 7 0 1 1 16 0 0 1 34 iii) By back-substitution we get x 3 = 34, x 2 = 16- 34 =- 18 and x 1 =- 7- (- 18) = 11. Hence the general solution is ( x 1 , x 2 , x 3 ) = (11 ,- 18 , 34). 1 2 b) x 1 + x 2 + 2 x 3 + x 4 = 3 x 1 + 2 x 2 + 4 x 3 + x 4 = 7 x 1 + x 4 =- 21 Solution: i) 1 1 2 1 3 1 2 4 1 7 1 0 0 1- 21 . ii) 1 1 2 1 3 1 2 4 1 7 1 0 0 1- 21 r 2- r 1 r 3- r 1 1 1 2 1 3 1 2 0 4- 1- 2 0- 24 r 3 + r 2 1 1 2 1 3 0 1 2 0 4 0 0 0 0- 20 (- 1 20 ) r 3 1 1 2 1 3 0 1 2 0 4 0 0 0 0 1 iii) Since the last row in REF has the form [0 | 1] the system is inconsistent....

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