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: 54

Answer

Inconsistent (no solution)

Work Step by Step

We are given the system of equations: $\begin{cases} 2x-3y-z=0\\ -x+2y+z=5\\ 3x-4y-z=1 \end{cases}$ Write the augmented matrix: $\begin{bmatrix} 2&-3&-1&|&0\\-1&2&1&|&5\\3&-4&-1&|&1\end{bmatrix}$ Perform row operations to bring the matrix to the reduced row echelon form: $R_1=r_2+r_1$ $\begin{bmatrix} 1&-1&0&|&5\\-1&2&1&|&5\\3&-4&-1&|&1\end{bmatrix}$ $R_2=r_1+r_2$ $\begin{bmatrix}1&-1&0&|&5\\0&1&1&|&10\\3&-4&-1&|&1\end{bmatrix}$ $R_3=-3r_1+r_3$ $\begin{bmatrix}1&-1&0&|&5\\0&1&1&|&10\\0&-1&-1&|&-14\end{bmatrix}$ $R_1=r_2+r_1$ $\begin{bmatrix}1&0&1&|&15\\0&1&1&|&10\\0&-1&-1&|&-14\end{bmatrix}$ $R_3=r_2+r_3$ $\begin{bmatrix}1&0&1&|&15\\0&1&1&|&10\\0&0&0&|&-4\end{bmatrix}$ As the last row contains only zeros to the left of the vertical bar and a nonzero element to its right, the system is inconsistent, and it has no solution.
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