Simplify. $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}} $$ $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{y}+\frac{1}{x}}=\square $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{y} + \frac{1}{x}}
$$

**Step 1: Simplify the Numerator**

$$
\frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy}
$$

**Step 2: Simplify the Denominator**

$$
\frac{1}{y} + \frac{1}{x} = \frac{x + y}{xy}
$$

**Step 3: Divide the Simplified Numerator by the Simplified Denominator**

$$
\frac{\frac{x^2 - y^2}{xy}}{\frac{x + y}{xy}} = \frac{x^2 - y^2}{xy} \times \frac{xy}{x + y} = \frac{x^2 - y^2}{x + y}
$$

**Step 4: Factor and Simplify Further**

$$
x^2 - y^2 = (x + y)(x - y)
$$

So,

$$
\frac{(x + y)(x - y)}{x + y} = x - y
$$

**Final Answer:**

$$
x - y
$$
$$ x - y $$

Simplify.