Let's start with the given equation:
[tex]\[ c = \frac{a}{a + b} \][/tex]
We need to isolate \( a \) in terms of \( b \) and \( c \).
Step 1: Multiply both sides by \( (a + b) \) to clear the fraction:
[tex]\[ c(a + b) = a \][/tex]
Step 2: Distribute \( c \) on the left-hand side:
[tex]\[ ca + cb = a \][/tex]
Step 3: Move all terms involving \( a \) to one side of the equation. Subtract \( a \) from both sides:
[tex]\[ ca - a = -cb \][/tex]
Step 4: Factor \( a \) out on the left-hand side:
[tex]\[ a(c - 1) = -cb \][/tex]
Step 5: Divide both sides by \( (c - 1) \) to isolate \( a \):
[tex]\[ a = \frac{-cb}{c - 1} \][/tex]
To simplify the expression, we can multiply the numerator and the denominator by \(-1\):
[tex]\[ a = \frac{cb}{1 - c} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{a = \frac{bc}{1 - c}} \][/tex]
The correct choice is:
(A) [tex]\(a = \frac{bc}{1 - c}\)[/tex]
