Answer :
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]
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]
