Express the rational expression in lowest terms. 3x-3y-bx+by/3x-3y+bx-by (The answer is 3-b/3+b I just need to see how the problem is done, thank you!)

Question

Express the rational expression in lowest terms. 3x-3y-bx+by/3x-3y+bx-by  (The answer is 3-b/3+b I just need to see how the problem is done, thank you!)

phi · Answer

$$ \frac{3x-3y-bx+by}{3x-3y+bx-by }$$

first do the top
it looks like we can factor 3 from the first 2 terms
3x-3y = 3(x-y)
also we can factor out a "b" from the last two terms:
-bx+by=  b(-x+y)
we have  (x-y)  in the first term and (-x+y) in the 2nd
but if we factor out a -1, we have  -(x-y)
so the top is
3(x-y) - b(x-y)

next, treat (x-y) as one "thing" and factor it out 
we get 
(3-b)(x-y)
now do something similar for the bottom