# How do you simplify (a^2-5a)/(3a-18)-(7a-36)/(3a-18)?

Mar 15, 2017

$\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18} = \frac{a}{3} - 2$

with exclusion $a \ne 6$

#### Explanation:

Since the denominators are identical, we can start by simply subtracting the numerators:

$\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18} = \frac{{a}^{2} - 5 a - 7 a + 36}{3 a - 18}$

$\textcolor{w h i t e}{\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18}} = \frac{{a}^{2} - 12 a + 36}{3 a - 18}$

$\textcolor{w h i t e}{\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18}} = \frac{\left(a - 6\right) \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{a - 6}}}\right)}{3 \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{a - 6}}}\right)}$

$\textcolor{w h i t e}{\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18}} = \frac{a - 6}{3}$

$\textcolor{w h i t e}{\frac{{a}^{2} - 5 a}{3 a - 18} - \frac{7 a - 36}{3 a - 18}} = \frac{a}{3} - 2$

with exclusion $a \ne 6$

Note that if $a = 6$, then the original expression is undefined, but the simplified one is defined. So we need to specify that $a = 6$ is excluded.