# How do you solve for u in 4/(u+6)=6/(u+6)+2?

Jan 11, 2018

$u = - 7$

#### Explanation:

First, we need to put the requirements, that is $u \ne - 6$ because then the denominator will be $0$, and make the equation undefined.

Then,

$\frac{4}{u + 6} = \frac{6}{u + 6} + 2$

$- \frac{2}{u + 6} = 2$

$u + 6 = - 1$

$u = - 7$

Feb 8, 2018

A different approach: $u = - 7$

#### Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\frac{4}{u + 6} = \frac{6}{u + 6} + 2 \textcolor{w h i t e}{\text{d")->color(white)("d}} \frac{4}{u + 6} = \frac{6}{u + 6} + \left[2 \textcolor{red}{\times 1}\right]}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddddddd")->color(white)("d}} \frac{4}{u + 6} = \frac{6}{u + 6} + \left[2 \textcolor{red}{\times \frac{u + 6}{u + 6}}\right]}$

color(green)(color(white)("dDDDDddddddddddd")->color(white)("d")4/(u+6)=6/(u+6)+color(white)("d")(2(u+6))/(u+6)

Now all the denominators (bottom numbers) are the same we can forget about them.

Or, as a purist would say: multiply all of both sides by $\left(u + 6\right)$. This cancels out the denominators which is THE SAME THING!

$\textcolor{g r e e n}{4 = 6 + 2 \left(u + 6\right)}$

$\textcolor{g r e e n}{4 = 6 + 2 u + 12}$

$\textcolor{g r e e n}{4 = 18 + 2 u}$

Subtract 18 from both sides

$\textcolor{g r e e n}{2 u = - 14}$

Divide both sides by 2

$\textcolor{g r e e n}{u = - 7}$

$\textcolor{b l u e}{\text{Foot note: as "u" can only take on one value for this to work }}$$\textcolor{b l u e}{\text{and this is not -6 then we do not need to state that } x \ne - 6}$