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 - 1.2 Gaussian Elimination and Gauss-Jordan Elimination - 1.2 Exercises: 31

Answer

$x=4$ $ y=-3$ $z=2 $

Work Step by Step

Write the augmented matrix of the system of linear equations. $ \begin{bmatrix} 1 &0 & -3 & -2\\ 3 & 1 & -2 & 5\\ 2 & 2 &1 & 4 \end{bmatrix} $ Add -3 times the 1st row to the 2nd row to produce a new 2nd row. Add -2 times the 1st row to the 3rd row to produce a new 3rd row. $ \begin{bmatrix} 1 &0 & -3 & -2\\ 0 & 1 & 7 & 11\\ 0 & 2 &7 &8 \end{bmatrix} $ Add -2 times the 2nd row to the 3rd row to produce a new 3rd row. $ \begin{bmatrix} 1 &0 & -3 & -2\\ 0 & 1 & 7 & 11\\ 0 & 0 &-7 & -14 \end{bmatrix} $ Divide the third row by -7. $ \begin{bmatrix} 1 &0 & -3 & -2\\ 0 & 1 & 7 & 11\\ 0 & 0 &1 & 2 \end{bmatrix} $ Use back-substitution to find the solution. $z=2 $ $y+7z=11 \rightarrow y=-3$ $x-3z=-2 \rightarrow x=4$
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