Let's fill in the blanks step by step.
Starting with the expression on the left:
- Substitute \( x = 6 \) and \( y = 4 \) into \( \frac{2}{3}x - \frac{3}{4}y - \frac{1}{6}x + \frac{1}{2}y \):
\[ \frac{2}{3}(6) - \frac{3}{4}(4) - \frac{1}{6}(6) + \frac{1}{2}(4) \]
Calculating each term:
- \( \frac{2}{3}(6) = 4 \)
- \( \frac{3}{4}(4) = 3 \)
- \( \frac{1}{6}(6) = 1 \)
- \( \frac{1}{2}(4) = 2 \)
So the left side becomes:
\[ 4 - 3 - 1 + 2 \]
Now evaluating this:
\[ 4 - 3 = 1 \] \[ 1 - 1 = 0 \] \[ 0 + 2 = 2 \]
So the left expression evaluates to 2.
Now for the right expression:
- Substitute \( x = 6 \) and \( y = 4 \) into \( \frac{1}{2}x - \frac{1}{2}y + \frac{1}{4}y \):
\[ \frac{1}{2}(6) - \frac{1}{2}(4) + \frac{1}{4}(4) \]
Calculating each term:
- \( \frac{1}{2}(6) = 3 \)
- \( \frac{1}{2}(4) = 2 \)
- \( \frac{1}{4}(4) = 1 \)
So the right side becomes:
\[ 3 - 2 + 1 \]
Now evaluating this:
\[ 3 - 2 = 1 \] \[ 1 + 1 = 2 \]
So the right expression also evaluates to 2.
Finally, comparing both sides:
The expressions are equivalent.
To summarize, we fill in the blanks as follows:
4 - 3 - 1 + 2
1
The expressions are equivalent.