To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we will combine like terms.
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Start by combining 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. \[ \frac{6}{7}c = \frac{12}{14}c \quad \text{and} \quad \frac{1}{2}c = \frac{7}{14}c \] Thus, \[ \frac{12}{14}c - \frac{7}{14}c = \frac{5}{14}c \]
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Now, combine the \( d \) terms: \[ -\frac{5}{9}d + \frac{1}{3}d \] We need a common denominator again. The least common multiple of 9 and 3 is 9. \[ \frac{1}{3}d = \frac{3}{9}d \] So, \[ -\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d \]
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Finally, combine the results: \[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \] The closest response among the given options would be: \[ 514c - 29d \] However, note that this suggests a misunderstanding of how to accurately represent your expression, and the final answer should more accurately state \( \frac{5}{14}c - \frac{2}{9}d \).