To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we will first combine like terms for \(c\) and \(d\).
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Combine 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. Thus: \[ \frac{6}{7} = \frac{6 \cdot 2}{7 \cdot 2} = \frac{12}{14} \] \[ \frac{1}{2} = \frac{1 \cdot 7}{2 \cdot 7} = \frac{7}{14} \] Now we can subtract: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{5}{14}c \]
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Combine 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. Thus: \[ \frac{1}{3} = \frac{1 \cdot 3}{3 \cdot 3} = \frac{3}{9} \] Now we can subtract: \[ -\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d \]
Putting it all together, we have: \[ \frac{5}{14}c - \frac{2}{9}d \]
So the final simplified result is: \[ \frac{5}{14}c - \frac{2}{9}d \]
The correct choice from the options provided is \(5/14c - 2/9d\).