To simplify the expression \( \frac{6}{7}c - \frac{5}{9}d - \frac{1}{2}c + \frac{1}{3}d \), we will first find a common denominator for the fractions.
- Identify the denominators: 7, 9, 2, 3.
- The least common multiple (LCM) of these numbers is 126.
- Rewrite each term with the common denominator of 126:
-
For \( \frac{6}{7}c \): \[ \frac{6}{7}c = \frac{6 \times 18}{7 \times 18}c = \frac{108}{126}c \]
-
For \( -\frac{5}{9}d \): \[ -\frac{5}{9}d = -\frac{5 \times 14}{9 \times 14}d = -\frac{70}{126}d \]
-
For \( -\frac{1}{2}c \): \[ -\frac{1}{2}c = -\frac{1 \times 63}{2 \times 63}c = -\frac{63}{126}c \]
-
For \( \frac{1}{3}d \): \[ \frac{1}{3}d = \frac{1 \times 42}{3 \times 42}d = \frac{42}{126}d \]
-
Now we can substitute back into the expression: \[ \frac{108}{126}c - \frac{63}{126}c - \frac{70}{126}d + \frac{42}{126}d \]
-
Combine the terms for \( c \): \[ \frac{108 - 63}{126}c = \frac{45}{126}c \]
-
Combine the terms for \( d \): \[ -\frac{70 - 42}{126}d = -\frac{28}{126}d \]
-
Rewrite the expression: \[ \frac{45}{126}c - \frac{28}{126}d \]
-
Lastly, we can simplify further: \[ \frac{45}{126}c - \frac{28}{126}d = \frac{15}{42}c - \frac{14}{63}d \] But where possible based on original simplification possibilities, it's best formatted as: \[ \frac{5}{14}c - \frac{2}{9}d \]
So the simplified form of the expression is: \[ \frac{5}{14}c - \frac{2}{9}d \]
To answer your question, the correct response is: Start Fraction 5 over 14 End Fraction c minus Start Fraction 2 over 9 End Fraction d