Answer
Solution set: $\{(-1,3,5)\}$
Work Step by Step
Start with the augmented matrix, row reduce to
reduced row echelon form (Gauss-Jordan.)
$\left[\begin{array}{llll}
2 & -3 & 5 & 14\\
4 & -1 & -2 & -17\\
-1 & -1 & 1 & 3
\end{array}\right]\rightarrow\left(\begin{array}{l}
+R_{3}.\\
-2R_{1}.\\
.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 6 & 17\\
0 & 5 & -12 & -45\\
-1 & -1 & 1 & 3
\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
.\\
+R_{1}.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 6 & 17\\
0 & 5 & -12 & -45\\
0 & -5 & 7 & 20
\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
.\\
+R_{2}.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 6 & 17\\
0 & 5 & -12 & -45\\
0 & 0 & -5 & -25
\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
.\\
\div(-5).
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 6 & 17\\
0 & 5 & -12 & -45\\
0 & 0 & 1 & 5
\end{array}\right]\rightarrow\left(\begin{array}{l}
-6R_{3}.\\
+12R_{3}.\\
.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 0 & -13\\
0 & 5 & 0 & 15\\
0 & 0 & 1 & 5
\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
\div 5.\\
.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & -4 & 0 & -13\\
0 & 1 & 0 & 3\\
0 & 0 & 1 & 5
\end{array}\right]\rightarrow\left(\begin{array}{l}
+4R_{2}.\\
.\\
.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & 0 & 0 & -1\\
0 & 1 & 0 & 3\\
0 & 0 & 1 & 5
\end{array}\right]$
Solution set: $\{(-1,3,5)\}$
