# How do you solve rational equations [(x-1) / (x-3)] = [2 /(x-3)]?

Jun 10, 2016

The given example has no solutions.

#### Explanation:

In solving rational equations we must first eliminate any values from the domain for which the denominator would be zero. (Since dividion by zero is illegal).

For the given example this means we must forbid the possibility of $x - 3 = 0 \rightarrow x = 3$

Once this is done, wee need to convert the rational expressions so all terms have the same denominator. For the given example this step is not needed since all terms already have the same denominator.

Once this is done, we can equate the numerators.
(Basically this means that if $\frac{{\text{expression"_1)/d=("expresssion}}_{2}}{d}$ and $d \ne 0$
then ${\text{expression"_1="expression}}_{2}$

For the given case this means
$\textcolor{w h i t e}{\text{XXX}} x - 1 = 2$ provided $x \ne 3$
but
$\textcolor{w h i t e}{\text{XXX}} x - 1 = 2 \rightarrow x = 3$
and therefore this equation has no solution.