Let's simplify the given expression step by step:
[tex]\[
\frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}}
\][/tex]
### Step 1: Simplify the numerator
The numerator of the expression is:
[tex]\[
\frac{1}{x} + \frac{1}{y}
\][/tex]
To combine the fractions, we need a common denominator. The common denominator between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is [tex]\(xy\)[/tex]:
[tex]\[
\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy}
\][/tex]
So, the simplified numerator is:
[tex]\[
\frac{y + x}{xy}
\][/tex]
### Step 2: Simplify the denominator
The denominator of the expression is:
[tex]\[
\frac{x}{y} - \frac{y}{x}
\][/tex]
To combine these fractions, we also need a common denominator. The common denominator between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] is [tex]\(xy\)[/tex]:
[tex]\[
\frac{x}{y} = \frac{x^2}{xy}, \quad \frac{y}{x} = \frac{y^2}{xy}
\][/tex]
Thus, the denominator becomes:
[tex]\[
\frac{x^2}{xy} - \frac{y^2}{xy} = \frac{x^2 - y^2}{xy}
\][/tex]
### Step 3: Combine the simplified numerator and denominator
Now we rewrite the original expression using the simplified numerator and denominator:
[tex]\[
\frac{\frac{y + x}{xy}}{\frac{x^2 - y^2}{xy}}
\][/tex]
This can be written as:
[tex]\[
\frac{y + x}{xy} \div \frac{x^2 - y^2}{xy}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{y + x}{xy} \times \frac{xy}{x^2 - y^2}
\][/tex]
Notice that [tex]\(xy\)[/tex] in the numerator of the second fraction and denominators cancel out:
[tex]\[
\frac{y + x}{\cancel{xy}} \times \frac{\cancel{xy}}{x^2 - y^2} = \frac{y + x}{x^2 - y^2}
\][/tex]
### Step 4: Simplify the resulting expression
Recognize that [tex]\(x^2 - y^2\)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - y^2 = (x + y)(x - y)
\][/tex]
Thus, our expression becomes:
[tex]\[
\frac{y + x}{(x + y)(x - y)}
\][/tex]
Since [tex]\(y + x\)[/tex] is the same as [tex]\(x + y\)[/tex], they cancel out:
[tex]\[
\frac{x + y}{(x + y)(x - y)} = \frac{1}{x - y}
\][/tex]
So the final simplified expression is:
[tex]\[
\boxed{\frac{1}{x - y}}
\][/tex]
