Answer
\( x = -2 \), \( y = 5 \), \( z = -2 \)
Work Step by Step
Taking our equations into matrix form:
\[
\begin{bmatrix}
1 & 1 & 0 & 3 \\
0 & 2 & 0 & 10 \\
3 & 2 & -4 & 12
\end{bmatrix}
\]
1. Divide \( \text{row 2} \) by 2:
\[
\begin{bmatrix}
1 & 1 & 0 & 3 \\
0 & 1 & 0 & 5 \\
3 & 2 & -4 & 12
\end{bmatrix}
\]
2. Subtract \( \text{row 2} \) from \( \text{row 1} \) and \( 2 \times \text{row 2} \) from \( \text{row 3} \):
\[
\begin{bmatrix}
1 & 0 & 0 & -2 \\
0 & 1 & 0 & 5 \\
3 & 0 & -4 & 2
\end{bmatrix}
\]
3. Subtract \( 3 \times \text{row 1} \) from \( \text{row 3} \):
\[
\begin{bmatrix}
1 & 0 & 0 & -2 \\
0 & 1 & 0 & 5 \\
0 & 0 & -4 & 8
\end{bmatrix}
\]
4. Divide \( \text{row 3} \) by \(-4\):
\[
\begin{bmatrix}
1 & 0 & 0 & -2 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & -2
\end{bmatrix}
\]
