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: 31


Consistent $\{(x,y,z)|x=-2z-1,y=4z-2,z\text{ any real number}\}$

Work Step by Step

We are given the reduced row echelon form of a system of linear equations: $\begin{bmatrix}1&0&2&|&-1\\0&1&-4&|&-2\\0&0&0&|&0\end{bmatrix}$ Write the system of equations corresponding to the given matrix: $\begin{cases} 1x+0y+2z=-1\\ 0x+1y-4z=-2\\ 0x+0y+0z=0 \end{cases}$ $\begin{cases} x+2z=-1\\ y-4z=-2\\ 0=0 \end{cases}$ 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 $z$: $y-4z=-2\Rightarrow y=4z-2$ $x+2z=-1\Rightarrow x=-2z-1$ The solution set is: $\{(x,y,z)|x=-2z-1,y=4z-2,z\text{ any real number}\}$
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