# How do you solve 1/5+1/x=1/2?

May 27, 2016

$x = \frac{10}{3}$

#### Explanation:

First, we eliminate the fractions from the equation by multiplying by the least common multiple of all denominators. As none of the denominators share any prime factors, the least common multiple is $5 \times x \times 2 = 10 x$

$\frac{1}{5} + \frac{1}{x} = \frac{1}{2}$

$\implies 10 x \left(\frac{1}{5} + \frac{1}{x}\right) = 10 x \left(\frac{1}{2}\right)$

$\implies \frac{10 x}{5} + \frac{10 x}{x} = \frac{10 x}{2}$

$\implies 2 x + 10 = 5 x$

$\implies 5 x - 2 x = 2 x + 10 - 2 x$

$\implies 3 x = 10$

$\implies \frac{3 x}{3} = \frac{10}{3}$

$\therefore x = \frac{10}{3}$

Finally, we check our answer by plugging in our result:

$\frac{1}{5} + \frac{1}{\frac{10}{3}} = \frac{1}{5} + \frac{3}{10}$

$= \frac{2}{10} + \frac{3}{10}$

$= \frac{5}{10}$

$= \frac{1}{2}$ as desired