How do you solve 1/2x-5/3=-1/2x+19/4?

Jan 25, 2017

See the entire solution process below:

Explanation:

First, multiply both sides of the equation by $\textcolor{red}{12}$ (the lowest common denominator of all the fractions) to eliminate the fractions while keeping the equation balanced:

$\textcolor{red}{12} \left(\frac{1}{2} x - \frac{5}{3}\right) = \textcolor{red}{12} \left(- \frac{1}{2} x + \frac{19}{4}\right)$

$\left(\textcolor{red}{12} \times \frac{1}{2} x\right) - \left(\textcolor{red}{12} \times \frac{5}{3}\right) = \left(\textcolor{red}{12} \times - \frac{1}{2} x\right) + \left(\textcolor{red}{12} \times \frac{19}{4}\right)$

$\left(6 \times 1 x\right) - \left(4 \times 5\right) = \left(6 \times - 1 x\right) + \left(3 \times 19\right)$

$6 x - 20 = - 6 x + 57$

Next, add $\textcolor{red}{6 x}$ and $\textcolor{b l u e}{20}$ to each side of the equation to isolate the $x$ term on the left side of the equation:

$6 x - 20 + \textcolor{red}{6 x} + \textcolor{b l u e}{20} = - 6 x + 57 + \textcolor{red}{6 x} + \textcolor{b l u e}{20}$

$6 x + \textcolor{red}{6 x} - 20 + \textcolor{b l u e}{20} = - 6 x + \textcolor{red}{6 x} + 57 + \textcolor{b l u e}{20}$

$12 x - 0 = 0 + 77$

$12 x = 77$

Now, divide each side by $\textcolor{red}{12}$ to solve for $x$ while keeping the equation balanced:

$\frac{12 x}{\textcolor{red}{12}} = \frac{77}{\textcolor{red}{12}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{12}}} x}{\cancel{\textcolor{red}{12}}} = \frac{77}{12}$

$x = \frac{77}{12}$

Jan 25, 2017

$x = \text{77"/"12}$

Explanation:

$\text{1"/2"x" - "5"/"3" = -"1"/2"x" + "19"/"4}$

$\text{1"/2"x" + "1"/2"x" = "19"/"4" + "5"/"3}$

$\text{1x + 1x"/"2" = "(19)(3) + (5)(4)"/"12}$

$\text{2x"/"2" = "57 + 20"/"12}$

$x = \text{77"/12}$