Question

Use the expansion shown to simplify the expression.

One-half (3 x minus two-thirds y) + 3 (StartFraction x over 2 EndFraction + one-half y)

Equals (one-half) (StartFraction 3 over 1 EndFraction x) + (one-half) (Negative two-thirds y) + (StartFraction 3 over 1 EndFraction) (one-half x) + (StartFraction 3 over 1 EndFraction) (one-half y)

Equals (three-halves x) + (negative one-third y) + (three-halves x) + (three-halves y)

Answer 1

Let's simplify the expression step-by-step using the expansion you've provided.

We start with the expression:

\[
\frac{1}{2} (3x - \frac{2}{3}y) + 3 \left(\frac{x}{2} + \frac{1}{2}y\right)
\]

Expanding this expression according to your breakdown, we have:

1. **Distributing \(\frac{1}{2}\) to the first part:**

\[
\frac{1}{2} (3x) + \frac{1}{2} (-\frac{2}{3}y) = \frac{3}{2}x - \frac{1}{3}y
\]

2. **Distributing \(3\) to the second part:**

\[
3 \left(\frac{x}{2}\right) + 3 \left(\frac{1}{2}y\right) = \frac{3}{2}x + \frac{3}{2}y
\]

Now, we can combine all these pieces together:

\[
\left(\frac{3}{2}x - \frac{1}{3}y\right) + \left(\frac{3}{2}x + \frac{3}{2}y\right)
\]

Combine the \(x\) terms and the \(y\) terms separately:

1. **Combining \(x\) terms:**

\[
\frac{3}{2}x + \frac{3}{2}x = 3x
\]

2. **Combining \(y\) terms:**

\[
-\frac{1}{3}y + \frac{3}{2}y
\]

To combine these \(y\) terms, we need a common denominator. The least common multiple of \(3\) and \(2\) is \(6\). So we can rewrite:

\[
-\frac{1}{3}y = -\frac{2}{6}y \quad \text{and} \quad \frac{3}{2}y = \frac{9}{6}y
\]

Now combine these:

\[
-\frac{2}{6}y + \frac{9}{6}y = \frac{7}{6}y
\]

Putting it all together, we arrive at the simplified expression:

\[
3x + \frac{7}{6}y
\]

Thus, the simplified version of the original expression is:

\[
3x + \frac{7}{6}y
\]