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: 33

Answer

Consistent Solution set: $\{(x_1,x_2,x_3,x_4)|x_1=1,x_2=2-x_4,x_3=3-2x_4,x_4\text{ is any real number}\}$

Work Step by Step

We are given the reduced row echelon form of a system of linear equations: $\begin{bmatrix}1&0&0&0&|&1\\0&1&0&1&|&2\\0&0&1&2&|&3\end{bmatrix}$ Write the system of equations corresponding to the given matrix: $\begin{cases} 1x_1+0x_2+0x_3+0x_4=1\\ 0x_1+1x_2+0x_3+1x_4=2\\ 0x_1+0x_2+1x_3+2x_4=3 \end{cases}$ $\begin{cases} x_1=1\\ x_2+x_4=2\\ x_3+2x_4=3 \end{cases}$ The system is consistent. As the number of equations is less than the number of variables, it has infinitely many solutions. Express $x_1,x_2,x_3$ in terms of $x_4$: $x_3+2x_4=3\Rightarrow x_3=3-2x_4$ $x_2+x_4=2\Rightarrow x_2=2-x_4$ $x_1=1$ The solution set is: $\{(x_1,x_2,x_3,x_4)|x_1=1,x_2=2-x_4,x_3=3-2x_4,x_4\text{ is any real number}\}$
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