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 first combine the terms that involve \( c \) and then combine the terms that involve \( d \).
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Combine \( c \) terms: \[ \frac{6}{7}c - \frac{1}{2}c \] To combine these fractions, we need a common denominator. The least common multiple of 7 and 2 is 14.
- Convert \( \frac{6}{7} \) to 14 as the denominator: \[ \frac{6}{7} = \frac{6 \times 2}{7 \times 2} = \frac{12}{14} \]
- Convert \( \frac{1}{2} \) to 14 as the denominator: \[ \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14} \]
- Now subtract the fractions: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{5}{14}c \]
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Combine \( d \) 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.
- Convert \( \frac{1}{3} \) to 9 as the denominator: \[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \]
- Now combine the fractions: \[ -\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d \]
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Combine the results: So now we have: \[ \frac{5}{14}c - \frac{2}{9}d \]
Therefore, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
From the options presented, the closest response to \( \frac{5}{14}c - \frac{2}{9}d \) is: 5/14c - 2/9d
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