# How do you solve 2/3r + 3/4 = 7/12?

Sep 22, 2017

1) Multiply both sides by the common denominator
2) Factorize to find common factors in numerator and denominator
3) Find r

#### Explanation:

It might be painful to have a lot of denominators in equations, but fear not! We can handle it quite easily by multiplying both sides in the equations by the common denominator.
We can always do the same operations in an equation, as long as we do the same on both sides.

The common denominator in this case is 12. (If you want an explanation to this, then notify me!)

This gives us:
$\frac{2}{3} r + \frac{3}{4} = \frac{7}{12}$
$12 \left(\frac{2}{3} r + \frac{3}{4}\right) = 12 \cdot \left(\frac{7}{12}\right)$ | multiply by 12 on both sides
$\frac{12 \cdot 2}{3} r + \frac{12 \cdot 3}{4} = \frac{\cancel{12} \cdot 7}{\cancel{12}}$

Then we factorize 12 into 3 * 4 to cross out common numbers in the numerator and denominator:

$\frac{3 \cdot 4 \cdot 2}{3} r + \frac{3 \cdot 4 \cdot 3}{4} = 7$
$\frac{\cancel{3} \cdot 4 \cdot 2}{\cancel{3}} r + \frac{3 \cdot \cancel{4} \cdot 3}{\cancel{4}} = 7$
$8 r + 9 = 7$
$8 r = 7 - 9$
$8 r = - 2$
$\frac{8 r}{8} = - \frac{2}{8}$ | divide both sides by 8

This gives us:

$r = - \frac{1}{4}$