Sure, let's work through the expression step-by-step to simplify [tex]\(\frac{a}{a-b}+\frac{b}{b-a}\)[/tex].
1. Observe the expression:
[tex]\[
\frac{a}{a-b} + \frac{b}{b-a}
\][/tex]
Notice that [tex]\( \frac{b}{b-a} \)[/tex] can be rewritten because [tex]\( b-a = -(a-b) \)[/tex]. Therefore:
[tex]\[
\frac{b}{b-a} = \frac{b}{-(a-b)} = -\frac{b}{a-b}
\][/tex]
2. Rewrite the expression using this identity:
[tex]\[
\frac{a}{a-b} - \frac{b}{a-b}
\][/tex]
3. Combine the terms over a common denominator:
[tex]\[
\frac{a - b}{a-b}
\][/tex]
4. Since the numerator and the denominator are identical:
[tex]\[
\frac{a-b}{a-b} = 1 \quad \text{(as long as \(a \neq b\))}
\][/tex]
Thus, the simplified form of the expression [tex]\(\frac{a}{a-b}+\frac{b}{b-a}\)[/tex] is:
[tex]\[
1
\][/tex]
