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: 32

Answer

Consistent Solution set: $\{(x,y,z)|x=4-4z,y=2-3z,z\text{ any real number}\}$

Work Step by Step

We are given the reduced row echelon form of a system of linear equations: $\begin{bmatrix}1&0&4&|&4\\0&1&3&|&2\\0&0&0&|&0\end{bmatrix}$ Write the system of equations corresponding to the given matrix: $\begin{cases} 1x+0y+4z=4\\ 0x+1y+3z=2\\ 0x+0y+0z=0 \end{cases}$ $\begin{cases} x+4z=4\\ y+3z=2\\ 0=0 \end{cases}$ Because the reduced row echelon form has a row with only zeros, the system is consistent, having infinitely many solutions. Express $x,y$ in terms of $z$: $y+3z=2\Rightarrow y=2-3z$ $x+4z=4\Rightarrow x=4-4z$ The solution set is: $\{(x,y,z)|x=4-4z,y=2-3z,z\text{ any real number}\}$
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