# How do you solve \frac { 3} { y - 1} + \frac { 1} { y } = \frac { 5} { 4}?

Oct 18, 2017

color(magenta)(y=1/5 or x=4

#### Explanation:

$\frac{3}{y - 1} + \frac{1}{y} = \frac{5}{4}$

$\therefore \frac{3 \left(4 y\right) + 4 \left(y - 1\right) = 5 \left(y - 1\right) \left(y\right)}{\left(y - 1\right) \left(y\right) \left(4\right)}$

multiply both sides by$\left(y - 1\right) \left(y \left(4\right)\right)$

$\therefore 12 y + 4 y - 4 = 5 {y}^{2} - 5 y$

$\therefore 5 {y}^{2} - 5 y = 16 y - 4$

$\therefore 5 {y}^{2} - 21 y + 4 = 0$

$\therefore \left(5 y - 1\right) \left(y - 4\right) = 0$

$\therefore 5 y = 1 \mathmr{and} y = 4$

:.color(magenta)(y=1/5 or y=4

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check:-

substitute color(magenta)(y= 0.2

$\frac{3}{\left(\textcolor{m a \ge n t a}{0.2}\right) - 1} + \frac{1}{\textcolor{m a \ge n t a}{0.2}} = \frac{5}{4}$

$\frac{3}{0.2} + \frac{1}{0.2} = 1.25$

$- 3.75 + 5 = 1.25$

color(magenta)(1.25=1.25

substitute color(magenta)(y=4

$\therefore \frac{3}{\left(\textcolor{m a \ge n t a}{4}\right) - 1} + 1 \left(\textcolor{m a \ge n t a}{4}\right) = \frac{5}{4}$

$\therefore \frac{3}{3} + \frac{1}{4} = \frac{5}{4}$

:.color(magenta)(1 1/4=1 1/4