# How do you solve \frac { 2} { 3} x - 5= \frac { x } { 4} - 10?

May 17, 2017

$x = - 12$

#### Explanation:

Let's solve this one step at a time.

First, we need to get rid of the pesky fractions. Since there are 2 fractions, let's find the LCM (Lowest common multiple). Both 3 and 4 goes in 12, so let's multiply the equation by 12.
$\frac{2}{3} \cdot \frac{12}{1} = \frac{24}{3} = 8$

$\frac{x}{4} \cdot \frac{12}{1} = \frac{12 x}{4} = 3 x$
Therefore,
$8 x - 60 = 3 x - 120$
To get $x$ on one side, let's add 60 to both sides.
$8 x = 3 x - 60$
Subtract $3 x$ from both sides to get the constant (-60) on its own.
$5 x = - 60$
Finally, divide both sides by 5 to isolate $x$
$x = - 12$

May 17, 2017

$x = - 12$

#### Explanation:

$\text{collect terms in x on the left side and numeric values on the}$
$\text{right side}$

$\text{Note } \frac{2}{3} x = \frac{2 x}{3}$

$\text{subtract " x/4" from both sides}$

$\Rightarrow \frac{2 x}{3} - \frac{x}{4} - 5 = \cancel{\frac{x}{4}} \cancel{- \frac{x}{4}} - 10$

$\text{add 5 to both sides}$

$\frac{2 x}{3} - \frac{x}{4} \cancel{- 5} \cancel{+ 5} = - 10 + 5$

$\Rightarrow \frac{2 x}{3} - \frac{x}{4} = - 5$

$\text{before subtracting the fractions we require them to}$
$\text{have a "color(blue)"common denominator}$

$\text{ the lowest common multiple of 3 and 4 is 12}$

$\Rightarrow \frac{2 x}{3} \times \frac{4}{4} - \frac{x}{4} \times \frac{3}{3} = - 5$

$\Rightarrow \frac{8 x}{12} - \frac{3 x}{12} = - 5$

$\Rightarrow \frac{5 x}{12} = - 5$

$\text{multiply both sides by 12}$

$\cancel{12} \times \frac{5 x}{\cancel{12}} = \left(- 5 \times 12\right)$

$\Rightarrow 5 x = - 60$

$\text{divide both sides by 5}$

$\frac{\cancel{5} x}{\cancel{5}} = \frac{- 60}{5}$

$\Rightarrow x = - 12$

$\textcolor{b l u e}{\text{As a check}}$

Substitute this value into the equation and if both sides are equal then it is the solution.

$\text{left side } = \left(\frac{2}{3} \times - 12\right) - 5 = - 8 - 5 = - 13$

$\text{right side } = \frac{- 12}{4} - 10 = - 3 - 10 = - 13$

$\Rightarrow x = - 12 \text{ is the solution}$