Answer
$ x=-4,y=2,z=\displaystyle \frac{5}{2}$
or $\displaystyle \left(-4,2,\frac{5}{2}\right)$
Work Step by Step
Method: Gauss-Jordan. Row reduce the augmented matrix:
$\left[\begin{array}{rrr|r}
1 & -2 & 4 & 2 \\
-3 & 5 & -2 & 17 \\
4 & -3 & 0 & -22 \end{array}\right]\qquad\left\{\begin{array}{l}
.\\
R_{2}=r_{2}+3r_{1}.\\
R_{3}=r_{3}-4r_{1}.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & -2 & 4 & 2 \\
0 & -1 & 10 & 23 \\
0 & 5 & -16 & -30 \end{array}\right]\qquad\left\{\begin{array}{l}
R_{1}=r_{1}-2r_{2}.\\
R_{2}=-r_{2}.\\
R_{3}=r_{3}+5r_{2}.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & 0 & -16 & -44 \\
0 & 1 & -10 & -23 \\
0 & 0 & 34 & 85 \end{array}\right]\qquad\left\{\begin{array}{l}
.\\
.\\
R_{3}=r_{3}\div 34.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & 0 & -16 & -44 \\
0 & 1 & -10 & -23 \\
0 & 0 & 1 & 5/2 \end{array}\right]\qquad\left\{\begin{array}{l}
R_{1}=r_{1}+16r_{3}.\\
R_{2}=r_{2}+10r_{3}.\\
.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
1 & 0 & 0 & -4 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 5/2 \end{array}\right]$
Solution: $ x=-4,y=2,z=\displaystyle \frac{5}{2}$
or $\displaystyle \left(-4,2,\frac{5}{2}\right)$
