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 like terms for \( c \) and \( d \).
-
Combine \( c \) 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 each to have a denominator of 14: \[ \frac{6}{7}c = \frac{12}{14}c \] \[ -\frac{1}{2}c = -\frac{7}{14}c \] Now we can combine them: \[ \frac{12}{14}c - \frac{7}{14}c = \frac{5}{14}c \]
-
Combine \( d \) terms: \[ -\frac{5}{9}d + \frac{1}{3}d \] The least common multiple of 9 and 3 is 9.
Convert \(\frac{1}{3}d\) to have a denominator of 9: \[ \frac{1}{3}d = \frac{3}{9}d \] Now combine: \[ -\frac{5}{9}d + \frac{3}{9}d = -\frac{2}{9}d \]
-
Final result: Combine the \( c \) and \( d \) results: \[ \frac{5}{14}c - \frac{2}{9}d \]
Thus, the simplified expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
This matches the option: \[ \text{Start Fraction 5 over 14 End Fraction c minus Start Fraction 2 over 9 End Fraction d} \] Therefore, the correct response is: \[ \frac{5}{14}c - \frac{2}{9}d \]