Let's simplify the given expression step by step:

\[
\frac{1}{9}(5x + 3y) - \frac{1}{3}(x - y)
\]

### Step 1: Distribute the fractions.

For the first term, \(\frac{1}{9}(5x + 3y)\), distribute \(\frac{1}{9}\) to both \(5x\) and \(3y\):

\[
\frac{1}{9}(5x) + \frac{1}{9}(3y) = \frac{5x}{9} + \frac{3y}{9}
\]
Simplify \(\frac{3y}{9}\) to \(\frac{y}{3}\):

\[
\frac{5x}{9} + \frac{y}{3}
\]

For the second term, \(\frac{1}{3}(x - y)\), distribute \(\frac{1}{3}\) to both \(x\) and \(-y\):

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

### Step 2: Combine the two parts.

Now we have:

\[
\left(\frac{5x}{9} + \frac{y}{3}\right) - \left(\frac{x}{3} - \frac{y}{3}\right)
\]

### Step 3: Simplify the terms.

First, let's rearrange and group the terms:
\[
\frac{5x}{9} + \frac{y}{3} - \frac{x}{3} + \frac{y}{3}
\]

Now combine the like terms for \(x\) and \(y\). We can rewrite \(\frac{x}{3}\) as \(\frac{3x}{9}\) to have a common denominator with \(\frac{5x}{9}\):

\[
\frac{5x}{9} - \frac{3x}{9} = \frac{2x}{9}
\]

For the \(y\) terms, \(\frac{y}{3} + \frac{y}{3} = \frac{2y}{3}\).

### Final Answer:

\[
\frac{2x}{9} + \frac{2y}{3}
\]

This is the simplified form of the given expression.

Do you want more details on any step? Here are five related questions:

1. How can we simplify expressions with different denominators?
2. How do we apply the distributive property with fractions?
3. What are common mistakes when combining like terms with fractions?
4. How can we factor the simplified expression \(\frac{2x}{9} + \frac{2y}{3}\)?
5. Can we solve for specific values of \(x\) and \(y\) if provided?

**Tip**: Always look for common denominators when adding or subtracting fractions to simplify the process.

Learn how to simplify the algebraic expression 1/9(5x + 3y) - 1/3(x - y) through the distributive property and combining like terms. This step-by-step solution covers important algebraic concepts such as working with fractions and distributing terms, suitable for Grades 8-10.

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