To find the value of \( b \) in the given system of linear equations, let's solve the system step-by-step.
The given system of equations is:
[tex]\[
\begin{aligned}
a - b + c &= -6 \quad \text{(1)} \\
b - c &= 5 \quad \text{(2)} \\
2a - 2c &= 4 \quad \text{(3)}
\end{aligned}
\][/tex]
We start by simplifying equation (3):
[tex]\[
2a - 2c = 4 \implies a - c = 2 \quad \text{(4)}
\][/tex]
Now we have:
[tex]\[
\begin{aligned}
a - b + c &= -6 \quad \text{(1)} \\
b - c &= 5 \quad \text{(2)} \\
a - c &= 2 \quad \text{(4)}
\end{aligned}
\][/tex]
From equation (2):
[tex]\[
b - c = 5 \implies b = c + 5 \quad \text{(5)}
\][/tex]
We will substitute \( b = c + 5 \) from equation (5) into equations (1) and (4).
First, substitute \( b = c + 5 \) into equation (1):
[tex]\[
a - (c + 5) + c = -6 \implies a - c - 5 + c = -6 \implies a - 5 = -6 \implies a = -1 \quad \text{(6)}
\][/tex]
Next, substitute \( a = -1 \) into equation (4):
[tex]\[
-1 - c = 2 \implies c = -1 - 2 \implies c = -3 \quad \text{(7)}
\][/tex]
Now, we substitute \( c = -3 \) back into equation (5) to find \( b \):
[tex]\[
b = c + 5 \implies b = -3 + 5 \implies b = 2
\][/tex]
Thus, the value of \( b \) is \( 2 \).
The correct answer is:
[tex]\[
\boxed{2}
\][/tex]
