Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 11 - Systems of Equations and Inequalities - 11.2 Systems of Linear Equations: Matrices - 11.2 Assess Your Understanding - Page 731: 57


Inconsistent (no solution)

Work Step by Step

We are given the system of equations: $\begin{cases} 2x-2y+3z=6\\ 4x-3y+2z=0\\ -2x+3y-7z=1 \end{cases}$ Write the augmented matrix: $\begin{bmatrix} 2&-2&3&|&6\\4&-3&2&|&0\\-2&3&-7&|&1\end{bmatrix}$ Perform row operations to bring the matrix to the reduced row echelon form: $R_3=r_1+r_3$ $\begin{bmatrix} 2&-2&3&|&6\\4&-3&2&|&0\\0&1&-4&|&7\end{bmatrix}$ $R_2=-2r_1+r_2$ $\begin{bmatrix}2&-2&3&|&6\\0&1&-4&|&-12\\0&1&-4&|&7\end{bmatrix}$ $R_1=\dfrac{1}{2}r_1$ $\begin{bmatrix}1&-1&\frac{3}{2}&|&3\\0&1&-4&|&-12\\0&1&-4&|&7\end{bmatrix}$ $R_3=-r_2+r_3$ $\begin{bmatrix}1&-1&\frac{3}{2}&|&3\\0&1&-4&|&-12\\0&0&0&|&19\end{bmatrix}$ As the last row contains only zeros at the left of the vertical bar and a nonzero element at its right side, the system is inconsistent; therefore it has no solution.
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