Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 1 - Systems of Linear Equations - Review Exercises: 29

Answer

$x=2$ $y=-3$ $z=3$

Work Step by Step

$-x+y+2z=1$ $2x+3y+z=-2$ $5x+4y+2z=4$ Write the augmented matrix of the system of linear equations. $ \begin{bmatrix} -1 & 1 & 2 & 1\\ 2 & 3 & 1 & -2\\ 5 & 4 & 2 & 4 \end{bmatrix} $ Multiply the first row by -1. $ \begin{bmatrix} 1 & -1 & -2 & -1\\ 2 & 3 & 1 & -2\\ 5 & 4 & 2 & 4 \end{bmatrix} $ Add -2 times the 1st row to the 2nd row to produce a new 2nd row. Add -5 times the 1st row to the 3rd row to produce a new 3rd row. $ \begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 5 & 5 & 0\\ 0 & 9 & 12 & 9 \end{bmatrix} $ Divide the second row by 5. $ \begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 1 & 1 & 0\\ 0 & 9 & 12 & 9 \end{bmatrix} $ Add -9 times the 2nd row to the 3rd row to produce a new 3rd row. $ \begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 1 & 1 & 0\\ 0 & 0 & 3 & 9 \end{bmatrix} $ Use back-substitution to find the solution. $3z=9 \rightarrow z=3$ $y+z=0 \rightarrow y+3=0 \rightarrow y=-3$ $x-y-2z=-1 \rightarrow x+3-6=-1 \rightarrow x=2$
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