Fill in the blanks!

Expressions with Multiple Variables
Is 2/3x - 3/4y - 1/6x + 1/2y equivalent to 1/2x - 1/2y + 1/4y ?
Substitute values for both x and y.
x=6 y=4
2/3 (6) - 3/4 (4) - 1/6 (6) + 1/2 (4)
4 - _ - 1 + 2
2

1/2 (6) - 1/2 (4) + 1/4 (4)
3-2+_
_

The expressions are _.

1 answer

Let's fill in the blanks step by step.

Starting with the expression on the left:

  1. 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:

  1. 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.