# How do you solve 2/3 = 2 - (5x-3)/(x-1)?

Mar 30, 2016

$x = \frac{5}{11}$

#### Explanation:

We have $\frac{2}{3} = \frac{2}{1} - \frac{5 x - 3}{x - 1}$

We need a common denominator to apply the subtraction on the right.

If we multiply a number by 1 we do not change its value. However, 1 can come in many forms. Examples: $\frac{3}{3} \text{ ; "(5b)/(5b)" ; } \frac{x - 1}{x - 1}$

Multiply $\frac{2}{1}$ by 1 but in the form of $1 = \frac{x - 1}{x - 1}$ giving

$\text{ } \frac{2}{3} = \left(\frac{2}{1} \times \frac{x - 1}{x - 1}\right) - \frac{5 x - 3}{x - 1}$

$\text{ } \frac{2}{3} = \frac{2 \left(x - 1\right) - \left(5 x - 3\right)}{x - 1}$

$\text{ } \frac{2}{3} = \frac{2 x - 2 - 5 x + 3}{x - 1}$

$\text{ } \frac{2}{3} = \frac{- 3 x + 1}{x - 1}$

$\text{ } 2 \left(x - 1\right) = 3 \left(- 3 x + 1\right)$

$\text{ } 2 x - 2 = - 9 x + 3$

$\text{ } 11 x = 5$

$\text{ } x = \frac{5}{11}$