To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we need to combine like terms (terms with \( c \) and \( d \)).
Step 1: Combine the \( c \) terms
First, we will combine \( \frac{6}{7}c \) and \( -\frac{1}{2}c \):
- The least common multiple (LCM) of 7 and 2 is 14.
- Convert \( \frac{6}{7} \) to a fraction with a denominator of 14: \[ \frac{6}{7} = \frac{6 \times 2}{7 \times 2} = \frac{12}{14} \]
- Convert \( -\frac{1}{2} \) to a fraction with a denominator of 14: \[ -\frac{1}{2} = -\frac{1 \times 7}{2 \times 7} = -\frac{7}{14} \]
Now combine: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{12 - 7}{14}c = \frac{5}{14}c \]
Step 2: Combine the \( d \) terms
Next, we will combine \( -\frac{5}{9}d \) and \( \frac{1}{3}d \):
- The least common multiple of 9 and 3 is 9.
- Convert \( \frac{1}{3}d \) to a fraction with a denominator of 9: \[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \]
Now combine: \[ -\frac{5}{9}d + \frac{3}{9}d = \frac{-5 + 3}{9}d = -\frac{2}{9}d \]
Final Expression
Combining both simplified terms, we get: \[ \frac{5}{14}c - \frac{2}{9}d \]
So the simplified expression is:
\[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the correct response is: \( \frac{5}{14}c - \frac{2}{9}d \).