To determine the value of [tex]\( x \)[/tex] in the given system of linear equations, we need to solve the system step by step. The system is:
[tex]\[
\begin{cases}
2x + 8y - z = -9 \\
y + z = 7 \\
-x - y + z = 8 \\
\end{cases}
\][/tex]
Step 1: Solve one of the equations for a variable. Let's start with the second equation:
[tex]\[
y + z = 7 \implies z = 7 - y
\][/tex]
Step 2: Substitute [tex]\( z = 7 - y \)[/tex] into the other two equations:
For the first equation:
[tex]\[
2x + 8y - (7 - y) = -9 \\
2x + 8y - 7 + y = -9 \\
2x + 9y - 7 = -9 \\
2x + 9y = -2 \quad \text{(1)}
\][/tex]
For the third equation:
[tex]\[
-x - y + (7 - y) = 8 \\
-x - y + 7 - y = 8 \\
-x - 2y + 7 = 8 \\
-x - 2y = 1 \quad \text{(2)}
\][/tex]
Now we have a simplified system of two equations:
[tex]\[
\begin{cases}
2x + 9y = -2 \quad \text{(1)} \\
-x - 2y = 1 \quad \text{(2)} \\
\end{cases}
\][/tex]
Step 3: Solve the simplified system. Start with the second equation:
[tex]\[
-x - 2y = 1 \implies x = -1 - 2y \quad \text{(3)}
\][/tex]
Step 4: Substitute [tex]\( x = -1 - 2y \)[/tex] from equation (3) into equation (1):
[tex]\[
2(-1 - 2y) + 9y = -2 \\
-2 - 4y + 9y = -2 \\
-2 + 5y = -2 \\
5y = 0 \\
y = 0
\][/tex]
Step 5: Substitute [tex]\( y = 0 \)[/tex] back into equation (3) to find [tex]\( x \)[/tex]:
[tex]\[
x = -1 - 2(0) \\
x = -1
\][/tex]
So, the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{-1}
\][/tex]
The correct answer is B. -1
