## Elementary Linear Algebra 7th Edition

$x=2$ $y=-3$ $z=3$
$-x+y+2z=1$ $2x+3y+z=-2$ $5x+4y+2z=4$ Write the augmented matrix of the system of linear equations. $\begin{bmatrix} -1 & 1 & 2 & 1\\ 2 & 3 & 1 & -2\\ 5 & 4 & 2 & 4 \end{bmatrix}$ Multiply the first row by -1. $\begin{bmatrix} 1 & -1 & -2 & -1\\ 2 & 3 & 1 & -2\\ 5 & 4 & 2 & 4 \end{bmatrix}$ Add -2 times the 1st row to the 2nd row to produce a new 2nd row. Add -5 times the 1st row to the 3rd row to produce a new 3rd row. $\begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 5 & 5 & 0\\ 0 & 9 & 12 & 9 \end{bmatrix}$ Divide the second row by 5. $\begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 1 & 1 & 0\\ 0 & 9 & 12 & 9 \end{bmatrix}$ Add -9 times the 2nd row to the 3rd row to produce a new 3rd row. $\begin{bmatrix} 1 & -1 & -2 & -1\\ 0 & 1 & 1 & 0\\ 0 & 0 & 3 & 9 \end{bmatrix}$ Use back-substitution to find the solution. $3z=9 \rightarrow z=3$ $y+z=0 \rightarrow y+3=0 \rightarrow y=-3$ $x-y-2z=-1 \rightarrow x+3-6=-1 \rightarrow x=2$