Question

Simplify the following:

Answer 1

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

$\dfrac{4}{a-3}\Big(\dfrac{a-5}{a-5}\Big) +\dfrac{9}{a-5}\Big(\dfrac{a-3}{a-3}\Big)\\\\=\dfrac{4(a-5)+9(a-3)}{(a-3)(a-5)}$

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

$\dfrac{13a-47}{a^2-8a+15}$