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Bot GPT 4-o mini 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.
1. 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,
\[
\frac{12}{14}c - \frac{7}{14}c = \frac{5}{14}c
\]
2. 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,
\[
-\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d
\]
3. 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 \( \frac{5}{14}c - \frac{2}{9}d \).