Certainly! Let's simplify the given expression step-by-step:
We start with the expression:
[tex]\[
\frac{x^{-1} - y^{-1}}{x - 1 + y - 1}
\][/tex]
### Step-by-Step Simplification:
1. Simplify the Exponents and the Denominator:
- We can rewrite [tex]\(x^{-1}\)[/tex] as [tex]\(\frac{1}{x}\)[/tex] and [tex]\(y^{-1}\)[/tex] as [tex]\(\frac{1}{y}\)[/tex]:
[tex]\[
\frac{\frac{1}{x} - \frac{1}{y}}{x - 1 + y - 1}
\][/tex]
- Combine like terms in the denominator:
[tex]\[
x - 1 + y - 1 = x + y - 2
\][/tex]
- So the expression now looks like this:
[tex]\[
\frac{\frac{1}{x} - \frac{1}{y}}{x + y - 2}
\][/tex]
2. Combine the Numerator into a Single Fraction:
- We need a common denominator for the fractions in the numerator:
[tex]\[
\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}
\][/tex]
So the expression becomes:
[tex]\[
\frac{\frac{y - x}{xy}}{x + y - 2}
\][/tex]
3. Simplify the Compound Fraction:
- Dividing by a fraction is the same as multiplying by its reciprocal. Hence:
[tex]\[
\frac{y - x}{xy} \div (x + y - 2) = \frac{y - x}{xy} \cdot \frac{1}{x + y - 2} = \frac{y - x}{xy(x + y - 2)}
\][/tex]
4. Simplify the Final Result:
- Notice that the numerator [tex]\(y - x\)[/tex] is the negation of [tex]\(x - y\)[/tex]. Therefore, we can write:
[tex]\[
\frac{y - x}{xy(x + y - 2)} \Rightarrow \frac{-(x - y)}{xy(x + y - 2)} = \frac{-1 \cdot (x - y)}{xy(x + y - 2)}
\][/tex]
So the simplified form of the given expression is:
[tex]\[
\frac{-1}{y} + \frac{1}{x}\text{ all over }x + y - 2
\][/tex]
Putting it all together, we get:
[tex]\[
\boxed{\frac{-1/y + 1/x}{x + y - 2}}
\][/tex]
