assign3_soln

assign3_soln - Math 136 Assignment 3 Solution 1. For each...

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Math 136 Assignment 3 Solution 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 2 + x 3 = 2 x 1 + x 2 + x 3 = 3 2 x 1 + 3 x 2 + 3 x 3 = 9 Solution: i) 0 1 1 2 1 1 1 3 2 3 3 9 . ii) 0 1 1 2 1 1 1 3 2 3 3 9 R 1 R 2 1 1 1 3 0 1 1 2 2 3 3 9 R 3 - 2 R 1 1 1 1 3 0 1 1 2 0 1 1 3 R 3 - R 2 1 1 1 3 0 1 1 2 0 0 0 1 iii) Since the last row is ± 0 0 0 | 1 ² the system is inconsistent. b) 2 x 1 + x 2 - x 3 = 6 x 1 - 2 x 2 - 2 x 3 = 1 - x 1 + 12 x 2 + 8 x 3 = 7 Solution: i) 2 1 - 1 6 1 - 2 - 2 1 - 1 12 8 7 .
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2 ii) 2 1 - 1 6 1 - 2 - 2 1 - 1 12 8 7 R 1 R 2 1 - 2 - 2 1 2 1 - 1 6 - 1 12 8 7 R 2 - 2 R 1 R 3 + R 1 1 - 2 - 2 1 0 5 3 4 0 10 6 8 R 3 - 2 R 2 1 - 2 - 2 1 0 5 3 4 0 0 0 0 iii) Observe that x 3 is a free variable, so we let x 3 = t R and back-substitution we get 5 x 2 = 4 - 3 x 3 = 4 5 - 3 5 t and x 1 = 1 + 2 x 2 + 2 x 3 = 1 + 2 ( 4 5 - 3 5 t ) + 2 t = 13 5 + 4 5 t . Hence the general solution is x 1 x 2 x 3 = 13 5 + 4 5 t 4 5 - 3 5 t t = 13 / 5 4 / 5 0 + t 4 / 5 - 3 / 5 1 . c)
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assign3_soln - Math 136 Assignment 3 Solution 1. For each...

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