Answer
Solution set: $\{ ( -1$ , $6$ , $3 ) \}$
Work Step by Step
Augmented matrix:
$\left[\begin{array}{lllll}
3 & 1 & -1 & | & 0\\
2 & 3 & -5 & | & 1\\
1 & -2 & 3 & | & -4
\end{array}\right]\left\{\begin{array}{l}
\leftrightarrow R_{3}.\\
.\\
.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & -4\\
2 & 3 & -5 & | & 1\\
3 & 1 & -1 & | & 0
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftarrow R_{2}-2R_{1}\\
\leftarrow R_{3}-3R_{1}.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & -4\\
0 & 7 & -11 & | & 9\\
0 & 7 & -10 & | & 12
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\leftarrow R_{3}-R_{2}.
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & -4\\
0 & 7 & -11 & | & 9\\
0 & 0 & 1 & | & 3
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
\Rightarrow c=3
\end{array}\right.$
Back substitute into row/equation 2,
$7b-11(3)=9$
$7b=9+33$
$7b=42$
$b=6$
Back substitute into row/equation 1:
$a-2(6)+3(3)=-4$
$a=-4+12-9=-1$
Solution set: $\{ ( -1$ , $6$ , $3 ) \}$
