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

May 2, 2018

First, let's clear the denominators:

$- 4 \times \left(5 - b\right) = \left(2 b + 2\right) \times 3$

$- 20 + 4 b = 6 b + 6$

$4 b = 6 b = 26$

$- 2 b = 26$

$b = - 13$

Let's check our work:

$\frac{5 + 13}{3} = 6$

$\frac{2 \times - 13 + 2}{- 4} = 6$

Yep! $b = - 13$

May 2, 2018

$b = - 13$

#### Explanation:

There is ONE fraction on each side of the equal sign, so you may cross-multiply.

$\frac{\textcolor{b l u e}{\left(5 - b\right)}}{\textcolor{red}{3}} = \frac{\textcolor{red}{\left(2 b + 2\right)}}{\textcolor{b l u e}{- 4}}$

$\textcolor{red}{6 b + 6} = \textcolor{b l u e}{- 20 + 4 b}$

$6 b - 4 b = - 20 - 6$

$2 b = - 26$

$b = - 13$

May 2, 2018

multiply both sides by each fraction's denominator, then solve for $b$, which would give you $b = - 13$

#### Explanation:

What we will do is multiply both sides by the denominator of each side, thereby eliminating the need for fractions.

Starting with the left hand side:

$\frac{5 - b}{\cancel{3}} \cancel{\textcolor{red}{\times 3}} = \frac{2 b + 2}{- 4} \textcolor{red}{\times 3}$

$5 - b = \frac{\left(2 b + 2\right) \textcolor{red}{\times 3}}{- 4}$

$5 - b = \frac{6 b + 6}{- 4}$

Now, we multiply both sides by the denominator of the right-hand side:

$\left(5 - b\right) \textcolor{red}{\times - 4} = \frac{6 b + 6}{\cancel{- 4}} \cancel{\textcolor{red}{\times - 4}}$

$- 20 + 4 b = 6 b + 6$

Now, we'll re-arrange the equation so all of the terms related to $b$ are on one side, and the constants are on the other:

$- 20 + \cancel{4 b} \textcolor{red}{- \cancel{4 b}} = 6 b + 6 \textcolor{red}{- 4 b}$

$- 20 = 2 b + 6$

$- 20 \textcolor{red}{- 6} = 2 b + \cancel{6} \textcolor{red}{- \cancel{6}}$

$- 26 = 2 b$

Finally, divide both sides by the constant $b$ is multiplied by:

$- \frac{26}{\textcolor{red}{2}} = \frac{\cancel{2} b}{\cancel{\textcolor{red}{2}}}$

$\textcolor{g r e e n}{b = - 13}$