Question

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 _.

Answer 1

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:

2. 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**.