To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we will group the terms with \( c \) together and the terms with \( d \) together:
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Group the \( c \) terms: \[ \frac{6}{7}c - \frac{1}{2}c \]
To combine these, we need a common denominator. The least common multiple of 7 and 2 is 14. Rewriting the fractions gives:
\[ \frac{6}{7} = \frac{12}{14}, \quad -\frac{1}{2} = -\frac{7}{14} \]
Now we can combine these: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{12 - 7}{14}c = \frac{5}{14}c \]
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Now group the \( d \) terms: \[ -\frac{5}{9}d + \frac{1}{3}d \]
Again, we need a common denominator. The least common multiple of 9 and 3 is 9. Rewriting the fractions gives:
\[ \frac{1}{3} = \frac{3}{9} \]
Now we can combine these: \[ -\frac{5}{9}d + \frac{3}{9}d = \frac{-5 + 3}{9}d = -\frac{2}{9}d \]
Now putting everything together:
\[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the simplified expression is:
\[ \frac{5}{14}c - \frac{2}{9}d \]