<s>[INST] Simplify $ \frac{b}{a-b}+ \frac{a}{b-a}$ I have a question regarding simplification of an algebraic expression. 
Here is the problem:
$ \frac{b}{ a - b} + \frac{a}{b-a}  $
The outcome is $ -1 $
Here is how I try to simplify it:


*

*Add fractions: $ \frac{ b(-a+b) + a(a-b)}{(a-b)(-a+b)} $

*FOIL: $ \frac{a^2 - 2ab + b^2}{(a-b)(-a+b)} $

*Apply Binomial Decomposition: $ \frac{(a-b)^2}{(a-b)(-a+b)} $

*Reduce fractions: $ \frac{\color{red}{(a-b)}(a-b)}{\color{red}{(a-b)}(-a+b)}$


The answer is: $ \frac{a-b}{-a+b} $
Now in the book the answer is, as previously mentioned is $-1$
How do I get my answer $ \frac{a-b}{-a+b} $ to become $-1$ ? 
Could you, please, explain the steps necessary to get that answer? 
Thank you
 [/INST] Notice that $-a+b=-(a-b)$. Thus: $\frac{a-b}{-a+b}=\frac{a-b}{-(a-b)}=-\frac{a-b}{a-b}=-1$.
You could also have done this in a much easier way:
$\frac{b}{a-b}+\frac{a}{b-a}=\frac{b}{a-b}-\frac{a}{-(b-a)}=\frac{b}{a-b}-\frac{a}{a-b}=\frac{b-a}{a-b}=-1$
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