Let's begin by isolating the variable \( b \) in the equation \( a = \frac{1}{2} (b - c) \). Here are the steps we'll take:
1. Start with the equation:
\[ a = \frac{1}{2} (b - c) \]
2. To eliminate the factor of \(\frac{1}{2}\) in front of the parenthesis, we need to multiply both sides of the equation by 2. This will give us:
\[ 2 \times a = 2 \times \frac{1}{2} (b - c) \]
\[ 2a = (b - c) \]
Notice that the \(2\) on the right side simplifies with the \(\frac{1}{2}\) factor, leaving us with just \(b - c\) on the right side.
3. Now we need to isolate \(b\) by adding \(c\) to both sides of the equation:
\[ 2a + c = (b - c) + c \]
\[ 2a + c = b \]
Now we have \( b \) as the subject of the equation, and it is written as:
\[ b = 2a + c \]
This matches option a) \( b = 2a + c \), which is the correct rearrangement to make \( b \) the subject of the equation.