Answer
x = 3
y = -2
z = 5
w = 0
Work Step by Step
$\begin{bmatrix}
3 & 2 & -1 & 1 & |0\\
1 & -1 & 4 & 2 & |25\\
-2 & 1 & 2 & -1 & |2\\
1 & 1 & 1 & 1 & |6\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & 1 & |6\\
1 & -1 & 4 & 2 & |25\\
-2 & 1 & 2 & -1 & |2\\
3 & 2 & -1 & 1 & |0\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & 1 & |6\\
0 & 2 & -3 & -1 & |-19\\
0 & 3 & 4 & 1 & |14\\
0 & -1 & -4 & -2 & |-18\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & 1 & |6\\
0 & -1 & -4 & -2 & |-18\\
0 & 3 & 4 & 1 & |14\\
0 & 2 & -3 & -1 & |-19\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & 1 & |6\\
0 & -1 & -4 & -2 & |-18\\
0 & 0 & -8 & -5 & |-40\\
0 & 0 & -11 & -5 & |-55\\
\end{bmatrix}$
Back-Substitution:
Z:
$8z + 5w = 40$
$\underline{-11z - 5w = -55}$
-3z = -15
z = 5
W:
-8(5) - 5w = -40
-40 - 5w = -40
-5w = 0
w = 0
Y:
-y - 4(5) - 2(0) = -18
-y - 20 = -18
-y = 2
y = -2
X:
x - 2 + 5 + 0 = 6
x + 3 = 6
x = 3
In Total:
x = 3
y = -2
z = 5
w = 0
