Answer
Solution set =
$\displaystyle \{(x,y,z)\ | \ x=\frac{1}{5}-\frac{1}{5}z,\ y=-\frac{6}{5}+\frac{1}{5}z, z\in \mathbb{R}\}$
Work Step by Step
Method: Gauss-Jordan. Row reduce the augmented matrix:
$\left[\begin{array}{rrr|r}
2 & -3 & 1 & 4 \\
-3 & 2 & -1 & -3 \\
0 & -5 & 1 & 6 \end{array}\right]\qquad\left\{\begin{array}{l}
R_{1}=2r_{1}+r_{2}.\\
R_{2}=3r_{1}+2r_{2}.\\
.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & -4 & 1 & 5 \\
0 & -5 & 1 & 6 \\
0 & -5 & 1 & 6 \end{array}\right]\qquad\left\{\begin{array}{l}
R_{1}=r_{1}-\frac{4}{5}r_{2}.\\
R_{2}=r_{2}\div(-5).\\
R_{3}=r_{3}-r_{2}.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & 0 & 1/5 & 1/5 \\
0 & 1 & -1/5 & -6/5 \\
0 & 0 & 0 & 0 \end{array}\right]\qquad$
The system is consistent (the last row represents 0=0, which is always satisfied).
Taking $z\in \mathbb{R}$ as a parameter, we have:
Eq.2 $\Rightarrow y=-\displaystyle \frac{6}{5}+\frac{1}{5}z,$
Eq.$1 \displaystyle \Rightarrow x=\frac{1}{5}-\frac{1}{5}z$
Solution set = $\displaystyle \{(x,y,z)\ | \ x=\frac{1}{5}-\frac{1}{5}z,\ y=-\frac{6}{5}+\frac{1}{5}z, z\in \mathbb{R}\}$
