To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we will first group the \(c\) terms and the \(d\) terms separately.
Step 1: Group the like terms.
The \(c\) terms: \[ \frac{6}{7}c - \frac{1}{2}c \]
The \(d\) terms: \[ -\frac{5}{9}d + \frac{1}{3}d \]
Step 2: Simplify the \(c\) terms.
To combine \(\frac{6}{7}c\) and \(-\frac{1}{2}c\), we need a common denominator. The least common multiple of 7 and 2 is 14.
Convert \(\frac{6}{7}\) to have a denominator of 14: \[ \frac{6}{7} = \frac{6 \times 2}{7 \times 2} = \frac{12}{14} \]
Convert \(-\frac{1}{2}\) to have a denominator of 14: \[ -\frac{1}{2} = \frac{-1 \times 7}{2 \times 7} = \frac{-7}{14} \]
Now, we can combine them: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{12 - 7}{14}c = \frac{5}{14}c \]
Step 3: Simplify the \(d\) terms.
To combine \(-\frac{5}{9}d\) and \(\frac{1}{3}d\), we also need a common denominator. The least common multiple of 9 and 3 is 9.
Convert \(\frac{1}{3}\) to have a denominator of 9: \[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \]
Now, combine them: \[ -\frac{5}{9}d + \frac{3}{9}d = \frac{-5 + 3}{9}d = \frac{-2}{9}d \]
Step 4: Combine the simplified terms.
Putting it all together, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the final answer is: \[ \boxed{\frac{5}{14}c - \frac{2}{9}d} \]