Question

Simplify the following:

Answer 1

First, we can bring both fractions under the common denominator (a - 3)(a - 5):

`(4)/(a-3)\Big((a-5)/(a-5)\Big) +(9)/(a-5)\Big((a-3)/(a-3)\Big)\\\\=(4(a-5)+9(a-3))/((a-3)(a-5))`

Looking at the numerator, we can distribute and collect like terms:

`4(a-5)+9(a-3)=4a-20+9a-27=13a-47`

and looking at the denominator, we can do the same:

`(a-3)(a-5)=(a-3)a+(a-3)(-5)\\=a^2-3a-5a+15\\=a^2-8a+15`

With these simplified expressions, the final fraction becomes

`(13a-47)/(a^2-8a+15)`