The given expression is:

\[
\frac{x - \frac{3y}{xy}}{\frac{1}{x} + \frac{1}{y}}
\]

To simplify this expression, let's break it down step by step.

### Step 1: Simplify the numerator
The numerator is:

\[
x - \frac{3y}{xy}
\]

We can simplify \( \frac{3y}{xy} \) as follows:

\[
\frac{3y}{xy} = \frac{3}{x}
\]

Thus, the numerator becomes:

\[
x - \frac{3}{x}
\]

### Step 2: Simplify the denominator
The denominator is:

\[
\frac{1}{x} + \frac{1}{y}
\]

The common denominator of \( \frac{1}{x} \) and \( \frac{1}{y} \) is \( xy \), so we rewrite it as:

\[
\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}
\]

### Step 3: Combine the simplified numerator and denominator

Now, we have:

\[
\frac{x - \frac{3}{x}}{\frac{x + y}{xy}}
\]

### Step 4: Multiply the numerator by the reciprocal of the denominator

To simplify this expression, multiply the numerator by the reciprocal of the denominator:

\[
= \left(x - \frac{3}{x}\right) \cdot \frac{xy}{x + y}
\]

Distribute \( xy \) to both terms inside the parentheses:

\[
= \frac{xy(x) - xy\left(\frac{3}{x}\right)}{x + y}
\]

Simplifying:

\[
= \frac{x^2y - 3y}{x + y}
\]

### Final Answer:
The simplified expression is:

\[
\frac{x^2y - 3y}{x + y}
\]

Let me know if you need further details or explanations!

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Here are 5 related questions to extend this topic:

1. How would this expression change if the denominator had different terms, such as \( \frac{1}{x} - \frac{1}{y} \)?
2. What happens if \( y = 0 \) in this simplified form?
3. How would you simplify similar expressions with higher degrees in the numerator and denominator?
4. Could this expression be simplified differently using factoring techniques?
5. How do these operations change if the terms are complex numbers?

**Tip**: Always check for possible common factors between the numerator and denominator when simplifying expressions.

This page provides a detailed, step-by-step solution to simplifying the rational expression x - 3y / xy / 1 / x + 1 / y. Learn how to manipulate fractions, combine terms, and apply algebraic concepts. Suitable for high school students in Grades 10-12.

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