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:
- Distributing \(\frac{1}{2}\) to the first part:
\[ \frac{1}{2} (3x) + \frac{1}{2} (-\frac{2}{3}y) = \frac{3}{2}x - \frac{1}{3}y \]
- 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:
- Combining \(x\) terms:
\[ \frac{3}{2}x + \frac{3}{2}x = 3x \]
- 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:
\[ 3x + \frac{7}{6}y \]