# How do you solve x/(2x-6)=2/(x-4)?

Apr 14, 2017

Restrict the domain so that the solutions do not cause division by zero in the original equation.
Multiply both sides by both denominators.
Check.

#### Explanation:

Given: $\frac{x}{2 x - 6} = \frac{2}{x - 4}$

Restrict the values of x so that any solutions that would cause division by zero are discarded:

x/(2x-6)=2/(x-4);x!=3,x!=4

Multiply both sides of the equation by $\left(2 x - 6\right) \left(x - 4\right)$

(2x-6)(x-4)x/(2x-6)=(2x-6)(x-4)2/(x-4);x!=3,x!=4

Please observe how the factors cancel:

cancel(2x-6)(x-4)x/cancel(2x-6)=(2x-6)cancel(x-4)2/cancel(x-4);x!=3,x!=4

Here is the equation with the cancelled factor removed:

(x-4)x=(2x-6)2;x!=3,x!=4

Use the distributive property on both sides:

x^2-4x=4x-12;x!=3,x!=4

We can put the quadratic into standard form by adding $12 - 4 x$ to both sides:

x^2-8x+12=0;x!=3,x!=4

This factors and the restrictions can be dropped:

$\left(x - 2\right) \left(x - 6\right) = 0$

$x = 2 \mathmr{and} x = 6$

Check:

$\frac{2}{2 \left(2\right) - 6} = \frac{2}{2 - 4}$
$\frac{6}{2 \left(6\right) - 6} = \frac{2}{6 - 4}$

$\frac{2}{- 2} = \frac{2}{- 2}$
$\frac{6}{6} = \frac{2}{2}$

$- 1 = - 1$
$1 = 1$

This checks.