To solve the system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
2x - y + z = -8 \\
x + y + z = -4 \\
3x - y - z = -4
\end{array}
\right.
\][/tex]
We will follow a systematic approach to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
### Step 1: Write down the system of equations
[tex]\[
\left\{
\begin{array}{l}
2x - y + z = -8 \quad \text{(1)} \\
x + y + z = -4 \quad \text{(2)} \\
3x - y - z = -4 \quad \text{(3)}
\end{array}
\right.
\][/tex]
### Step 2: Add Equation (1) and Equation (3) to eliminate [tex]\( y \)[/tex]
[tex]\[
2x - y + z + 3x - y - z = -8 - 4
\][/tex]
[tex]\[
5x - 2y = -12 \quad \text{(4)}
\][/tex]
### Step 3: Add Equation (1) and Equation (2) to eliminate [tex]\( z \)[/tex]
[tex]\[
2x - y + z + x + y + z = -8 - 4
\][/tex]
[tex]\[
3x + 2z = -12 \quad \text{(5)}
\][/tex]
### Step 4: Add Equation (2) and Equation (3) to eliminate [tex]\( y \)[/tex]
[tex]\[
x + y + z + 3x - y - z = -4 - 4
\][/tex]
[tex]\[
4x = -8
\][/tex]
[tex]\[
x = -2 \quad \text{(6)}
\][/tex]
### Step 5: Substitute [tex]\( x = -2 \)[/tex] back into Equation (4) and Equation (5)
#### Substitute [tex]\( x = -2 \)[/tex] in Equation (4):
[tex]\[
5(-2) - 2y = -12
\][/tex]
[tex]\[
-10 - 2y = -12
\][/tex]
[tex]\[
-2y = -2
\][/tex]
[tex]\[
y = 1 \quad \text{(7)}
\][/tex]
#### Substitute [tex]\( x = -2 \)[/tex] in Equation (5):
[tex]\[
3(-2) + 2z = -12
\][/tex]
[tex]\[
-6 + 2z = -12
\][/tex]
[tex]\[
2z = -6
\][/tex]
[tex]\[
z = -3 \quad \text{(8)}
\][/tex]
### Step 6: Summarize the solution
The solution to the system of equations is:
[tex]\[
x = -2, \quad y = 1, \quad z = -3
\][/tex]
Thus, the correct answer is:
[tex]\[
(-2, 1, -3)
\][/tex]
