# How do you solve 5-2/(x-6) = (10-2x)/(x-6)?

May 29, 2017

no solution

#### Explanation:

Let's give everything the same denominator. Fortunately, two of the three components already share the same denominator $\left(x - 6\right)$.

If we multiply $5$ by $\frac{x - 6}{x - 6}$, all the components will be "combinable":

$\frac{5 x - 30}{x - 6} - \frac{2}{x - 6} = \frac{10 - 2 x}{x - 6}$

Combine like-terms

$\frac{\left(5 x - 30\right) - \left(2\right)}{x - 6} = \frac{10 - 2 x}{x - 6}$

Multiply by ($x - 6$) on both sides

$5 x - 30 - 2 = 10 - 2 x$

Simplify

$5 x - 32 = 10 - 2 x$

$7 x - 32 = 10$

Add $32$ to both sides

$7 x = 42$

Divide by $7$ on both sides

$x = 6$

Just to check our work, let's solve the equation, replacing $x$ with $6$:

$5 - \frac{2}{6 - 6} = \frac{10 - 2 \times 6}{6 - 6}$

$5 - \frac{2}{0} = \frac{- 2}{0}$

Uh oh! We're dividing by ZERO! that's illegal, so there are no solutions.