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 can group the like terms (the terms involving \( c \) and the terms involving \( d \)).
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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 \( 14 \).
- Convert \( \frac{6}{7}c \): \[ \frac{6}{7} = \frac{6 \times 2}{7 \times 2} = \frac{12}{14} \]
- Convert \( \frac{1}{2}c \): \[ \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14} \] Now combine: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{12 - 7}{14}c = \frac{5}{14}c \]
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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 \( 9 \).
- Convert \( \frac{1}{3}d \): \[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \] Now combine: \[ -\frac{5}{9}d + \frac{3}{9}d = \frac{-5 + 3}{9}d = \frac{-2}{9}d \]
Putting it all together, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
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