# How do you solve for x in (3/x) + (2/m) = (1/n) ?

Apr 23, 2016

$\frac{3}{x} = \frac{1}{n} - \frac{2}{m} = \frac{m - 2 n}{m n}$
$\implies \frac{x}{3} = \frac{m n}{m - 2 n}$
$: . x = \frac{3 m n}{m - 2 n}$

Apr 23, 2016

Manipulation shown in a lot of detail. Skip over the bits you know.

$x = \frac{3 m n}{m - 2 n}$

#### Explanation:

Tricks for manipulating this equation type.

$\textcolor{red}{\text{Very important "->" What you do to one side of the equation you do to the other side.}}$

$\textcolor{g r e e n}{\text{To move an add or subtract to the other side of the = turn it into 0}}$
$\textcolor{g r e e n}{\text{To move a multiply or divide to the other side of the = turn it into 1}}$
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Given:$\text{ } \left(\frac{3}{x}\right) + \left(\frac{2}{m}\right) = \left(\frac{1}{n}\right)$

Write as:
$\text{ } \frac{3}{x} + \frac{2}{m} = \frac{1}{n}$

We need to determine the value of $x$ so we need to 'get rid' of $\frac{2}{m}$ so that we have $\frac{3}{x}$ on its own.

Subtract color(blue)(2/m) " "underline("from both sides")

" "color(brown)(3/x+2/m color(blue)(-2/m)" "=" "1/n color(blue)(-2/m))
$\textcolor{w h i t e}{.}$

$\text{ } \frac{3}{x} + 0 = \frac{1}{n} - \frac{2}{m}$

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To do the next bit we need to combine $\frac{1}{n} - \frac{2}{m}$

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

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

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

$\text{ } \frac{3}{x} = \left(\frac{m}{m n}\right) - \left(\frac{2 n}{m n}\right)$

$\text{ } \frac{3}{x} = \frac{m - 2 n}{m n}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now you can turn everything upside down

$\text{ } \frac{x}{3} = \frac{m n}{m - 2 n}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Multiply both sides by $\textcolor{b l u e}{\text{ } 3}$

$\textcolor{b r o w n}{\text{ } x \times \frac{\textcolor{b l u e}{3}}{3} = \frac{\textcolor{b l u e}{3} m n}{m - 2 n}}$

But $\frac{3}{3} = 1$ giving:

$\text{ } x = \frac{3 m n}{m - 2 n}$