# How do you solve the equation (x/(x-1)) + (x/3) = (5/(x-1))?

Mar 13, 2018

$x = 2.594$ or $1.927$

#### Explanation:

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

We need a common denominator of $\left(x - 1\right) \times 3$

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

$\frac{3 x}{3 x - 3} + \frac{{x}^{2} - x}{3 x - 3} = \frac{15}{3 x - 3}$

$\frac{3 {x}^{2} + 3 x - x}{3 x - 3} = \frac{15}{3 x - 3}$

$3 {x}^{2} + 2 x = 15$

$3 {x}^{2} + 2 x - 15 = 0$

$\textcolor{g r e e n}{a} = \textcolor{g r e e n}{3}$
$\textcolor{b l u e}{b} = \textcolor{b l u e}{2}$
$\textcolor{\mathmr{and} a n \ge}{c} = \textcolor{\mathmr{and} a n \ge}{- 15}$

$- \frac{\textcolor{b l u e}{b}}{2 \times \textcolor{g r e e n}{a}} \pm \frac{\sqrt{{\textcolor{b l u e}{b}}^{2} - 4 \times \textcolor{g r e e n}{a} \times \textcolor{\mathmr{and} a n \ge}{c}}}{2 \times \textcolor{g r e e n}{a}}$

$- \frac{\textcolor{b l u e}{2}}{2 \times \textcolor{g r e e n}{3}} \pm \frac{\sqrt{{\textcolor{b l u e}{2}}^{2} - 4 \times \textcolor{g r e e n}{3} \times \textcolor{\mathmr{and} a n \ge}{15}}}{2 \times \textcolor{g r e e n}{3}}$

$- \frac{1}{3} \pm \frac{\sqrt{4 + 180}}{6}$

$\frac{1}{3} \pm \frac{\sqrt{184}}{6}$

Let's simplify the Square Root

$184 = 2 \times 2 \times 2 \times 23 = {2}^{2} \times 46$

$\sqrt{184} = \sqrt{{2}^{2}} \times \sqrt{46} = 2 \times \sqrt{46}$

$- \frac{1}{3} \pm \frac{2 \sqrt{46}}{6}$

$- \frac{1}{3} \pm \frac{\sqrt{46}}{3}$

So $x = 2.594$ or $1.927$