To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we can combine the like terms for \( c \) and \( d \) separately.
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Combine the \( c \) terms: \[ \frac{6}{7}c - \frac{1}{2}c \] To combine these fractions, we'll need a common denominator. The least common multiple of 7 and 2 is 14: \[ \frac{6}{7} = \frac{6 \times 2}{7 \times 2} = \frac{12}{14} \] \[ \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14} \] 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: \[ \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \] Now we can combine: \[ -\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d \]
Putting it all together: \[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
From the provided responses, it looks like the correct answer is: Start Fraction 5 over 14 End Fraction c minus Start Fraction 2 over 9 End Fraction d.