Answer
$(x,y,z)=(\frac{9}{2},\frac{5}{2},-2)$.
Work Step by Step
The augmented matrix is
$\begin{bmatrix}
1&-1 &0 & 2\\
0 & 2 & 1 & 3\\
0& 0 & 1 & -2
\end{bmatrix}$.
For the linear equation.
First column $=$ coefficients of $x$.
Second column $=$ coefficients of $y$
Third column $=$ coefficients of $z$
and fourth column $=$ the right-hand side.
The given matrix is a upper triangular form.
Rewrite the third row as a linear equation.
$\Rightarrow 0x+0y+1z=-2$
$\Rightarrow z=-2$
Rewrite the second row as a linear equation.
$\Rightarrow 0x+2y+z=3$
$\Rightarrow 2y+z=3$
Substitute $z=-2$.
$\Rightarrow 2y+(-2)=3$
Clear the parentheses.
$\Rightarrow 2y-2=3$
Add $2$ to both sides.
$\Rightarrow 2y-2+2=3+2$
Simplify.
$\Rightarrow 2y=5$
Divide both sides by $2$.
$\Rightarrow \frac{2y}{2}=\frac{5}{2}$
Simplify.
$\Rightarrow y=\frac{5}{2}$
Rewrite the first row as a linear equation.
$\Rightarrow 1x-1y+0z=2$
$\Rightarrow x-y=2$
Substitute $y=\frac{5}{2}$.
$\Rightarrow x-\frac{5}{2}=2$
Add $\frac{5}{2}$ to both sides.
$\Rightarrow x-\frac{5}{2}+\frac{5}{2}=2+\frac{5}{2}$
Simplify.
$\Rightarrow x=\frac{4+5}{2}$
$\Rightarrow x=\frac{9}{2}$
The solution set is $(x,y,z)=(\frac{9}{2},\frac{5}{2},-2)$.
