# How do you combine (2x)/(x-3) - (3x)/(x-5)?

May 24, 2015

As with adding or subtracting numerical fractions, the first step is to form a common denominator (bottom)...

$\frac{2 x}{x - 3} - \frac{3 x}{x - 5}$

$= \frac{2 x \left(x - 5\right)}{\left(x - 3\right) \left(x - 5\right)} - \frac{3 x \left(x - 3\right)}{\left(x - 3\right) \left(x - 5\right)}$

$= \frac{2 x \left(x - 5\right) - 3 x \left(x - 3\right)}{\left(x - 3\right) \left(x - 5\right)}$

$= \frac{2 {x}^{2} - 10 x - 3 {x}^{2} + 9 x}{\left(x - 3\right) \left(x - 5\right)}$

$= \frac{- {x}^{2} - x}{\left(x - 3\right) \left(x - 5\right)}$

$= - \frac{x \left(x + 1\right)}{\left(x - 3\right) \left(x - 5\right)}$

or if you prefer

$= - \frac{{x}^{2} + x}{{x}^{2} - 8 x + 15}$