# How do you solve  2/3+1/5=1/x?

May 12, 2016

$x = \frac{13}{2} \text{ " ->" } 6 \frac{1}{2}$

#### Explanation:

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

We need to find $x$ and one very cool trick in the mathematical tool box is that you can turn every thing upside down and still maintain the truth of the equation.

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

Multiply $\frac{5}{1}$ by 1 but in the form of $1 = \frac{2}{2}$

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

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

$\frac{13}{2} = x$

May 12, 2016

x = 15/13

#### Explanation:

There are 2 methods one ca use: One involves adding the two fractions to get one answer, and then inverting the whole equation.

The other involves multiplying by the LCM of the denominators to remove them altogether, and then solving for $x$.

Method 1:
$\frac{2}{3} + \frac{1}{5} = \frac{1}{x} \text{ "rArr (10+3)/15 = 1/x " } \Rightarrow \frac{13}{15} = \frac{1}{x}$

Inverting the equation gives $\frac{15}{13} = \frac{x}{1} \text{ } x = \frac{15}{13}$

Method 2: multiply through by the LCM ( $15 x$) to cancel the denominators:

$15 x \times \frac{2}{3} + 15 x \times \frac{1}{5} = 15 x \times \frac{1}{x}$

$5 x \times 2 + 3 x \times 1 = 15$

$10 x + 3 x = 15 \Rightarrow 13 x = 15 \text{ } \Rightarrow x = \frac{15}{13}$