# How do you solve 2/3x+5/6=7/8x-1/2?

Apr 29, 2017

See the entire solution process below:

#### Explanation:

First, multiply both sides of the equation by $\textcolor{red}{24}$ to eliminate the fractions while keeping the equation balanced $\textcolor{red}{24}$ is the Lowest Common Denominator of the 4 fractions:

$\textcolor{red}{24} \left(\frac{2}{3} x + \frac{5}{6}\right) = \textcolor{red}{24} \left(\frac{7}{8} x - \frac{1}{2}\right)$

$\left(\textcolor{red}{24} \cdot \frac{2}{3} x\right) + \left(\textcolor{red}{24} \cdot \frac{5}{6}\right) = \left(\textcolor{red}{24} \cdot \frac{7}{8} x\right) - \left(\textcolor{red}{24} \cdot \frac{1}{2}\right)$

$\left(\cancel{\textcolor{red}{24}} 8 \cdot \frac{2}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} x\right) + \left(\cancel{\textcolor{red}{24}} 4 \cdot \frac{5}{\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}}}\right) = \left(\cancel{\textcolor{red}{24}} 3 \cdot \frac{7}{\textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}} x\right) - \left(\cancel{\textcolor{red}{24}} 12 \cdot \frac{1}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}\right)$

$16 x + 20 = 21 x - 12$

Next, subtract $\textcolor{red}{16 x}$ and add $\textcolor{b l u e}{12}$ to each side of the equation to isolate the $x$ term while keeping the equation balanced:

$- \textcolor{red}{16 x} + 16 x + 20 + \textcolor{b l u e}{12} = - \textcolor{red}{16 x} + 21 x - 12 + \textcolor{b l u e}{12}$

$0 + 32 = \left(- \textcolor{red}{16} + 21\right) x - 0$

$32 = 5 x$

Now, divide each side of the equation by $\textcolor{red}{5}$ to solve for $x$ while keeping the equation balanced:

$\frac{32}{\textcolor{red}{5}} = \frac{5 x}{\textcolor{red}{5}}$

$\frac{32}{5} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} x}{\cancel{\textcolor{red}{5}}}$

$\frac{32}{5} = x$

$x = \frac{32}{5}$