To find the solution to the system of equations:
[tex]\[
\left\{\begin{array}{rcl}
-3x - y + z & = & 6 \quad \text{(1)} \\
-3x - y + 3z & = & 0 \quad \text{(2)} \\
x - 3z & = & 3 \quad \text{(3)}
\end{array}\right.
\][/tex]
We'll solve the system of equations step-by-step:
1. Start with equation (3) to solve for [tex]\(x\)[/tex] in terms of [tex]\(z\)[/tex]:
[tex]\[
x - 3z = 3 \quad \Rightarrow \quad x = 3 + 3z \quad \text{(4)}
\][/tex]
2. Substitute [tex]\(x = 3 + 3z\)[/tex] into equations (1) and (2):
- For equation (1):
[tex]\[
-3(3 + 3z) - y + z = 6 \\
-9 - 9z - y + z = 6 \\
-9 - 8z - y = 6 \\
-8z - y = 15 \\
y = -8z - 15 \quad \text{(5)}
\][/tex]
- For equation (2):
[tex]\[
-3(3 + 3z) - y + 3z = 0 \\
-9 - 9z - y + 3z = 0 \\
-9 - 6z - y = 0 \\
y = -6z - 9 \quad \text{(6)}
\][/tex]
3. Set equations (5) and (6) equal to each other, since they both equal [tex]\(y\)[/tex]:
[tex]\[
-8z - 15 = -6z - 9 \\
-8z + 6z = -9 + 15 \\
-2z = 6 \\
z = -3
\][/tex]
4. Substitute [tex]\(z = -3\)[/tex] back into equations (4) and (5):
- For [tex]\(x\)[/tex]:
[tex]\[
x = 3 + 3(-3) \\
x = 3 - 9 \\
x = -6
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
y = -8(-3) - 15 \\
y = 24 - 15 \\
y = 9
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = -6, y = 9, z = -3}
\][/tex]
