Answer
x = -1
y = 0
z = 6
w = 4
Work Step by Step
$\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
3 & -2 & 1 & -4 & |-13\\
-4 & 3 & -2 & 1 & |-4\\
-2 & 1 & -4 & 3 & |-10\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
0 & 10 & -8 & 2 & |-40\\
0 & -13 & 10 & -7 & |32\\
0 & -7 & 2 & -1 & |8\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
0 & 5 & -4 & 1 & |-20\\
0 & -13 & 10 & -7 & |32\\
0 & -7 & 2 & -1 & |8\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
0 & 5 & -4 & 1 & |-20\\
0 & 0 & 1 & 11 & |50\\
0 & 0 & -18 & 2 & |-100\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
0 & 5 & -4 & 1 & |-20\\
0 & 0 & 1 & 11 & |50\\
0 & 0 & -9 & 1 & |-50\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & -4 & 3 & -2 & |9\\
0 & 5 & -4 & 1 & |-20\\
0 & 0 & 1 & 11 & |50\\
0 & 0 & 0 & 100 & |400\\
\end{bmatrix}$
Back-Substitution:
W:
100w = 400
w = 4
Z:
z + 4(11) = 50
z + 44 = 50
z = 6
Y:
5y - 4(6) + 4 = -20
5y - 24 + 4 = -20
5y - 20 = -20
5y = 0
y = 0
X:
x - 4(0) + 3(6) - 2(4) = 9
x + 18 - 8 = 9
x + 10 = 9
x = -1
In Total:
x = -1
y = 0
z = 6
w = 4
