# How do you solve \frac { x - 1} { 3} + \frac { 2x + 1} { 5} = \frac { 3x - 1} { 4}?

Dec 18, 2016

$x = 7$

#### Explanation:

To solve this the first thing to do is make the equation simpler by eliminating the fractions. We can do this by multiplying each side of the equation by a common denominator, in this case $3 \cdot 5 \cdot 4 = 60$. This will eliminate the fractions and keep the equation balanced:

$\textcolor{red}{60} \left(\frac{\left(x - 1\right)}{3} + \frac{\left(2 x + 1\right)}{5}\right) = \textcolor{red}{60} \frac{\left(3 x - 1\right)}{4}$

$\left({\cancel{60}}^{20} \cdot \frac{x - 1}{\cancel{3}}\right) + \left({\cancel{60}}^{12} \cdot \frac{\left(2 x + 1\right)}{5}\right) = {\cancel{60}}^{15} \frac{\left(3 x - 1\right)}{4}$

$20 \left(x - 1\right) + 12 \left(2 x + 1\right) = 15 \left(3 x - 1\right)$

We can now expand the terms within parenthesis, group and combine like terms on each side of the equation:

$20 x - 20 + 24 x + 12 = 45 x - 15$

$20 x + 24 x - 20 + 12 = 45 x - 15$

$\left(20 + 24\right) x - 8 = 45 x - 15$

$44 x - 8 = 45 x - 15$

Now we can isolate the $x$ terms on one side of the equation and the constants on the other side of the equation:

$44 x - 8 \textcolor{red}{- 44 x + 15} = 45 x - 15 \textcolor{red}{- 44 x + 15}$

$44 x - 44 x - 8 + 15 = 45 x - 44 x - 15 + 15$

$0 - 8 + 15 = 45 x - 44 x - 0$

$- 8 + 15 = 45 x - 44 x$

$7 = \left(45 - 44\right) x$

$7 = x$

$x = 7$

Jan 1, 2017

$x = 7$

#### Explanation:

$\frac{x - 1}{3} + \frac{2 x + 1}{5} = \frac{3 x - 1}{4}$

multiply both sides by 3

$x - 1 + 3 \frac{2 x + 1}{5} = 3 \frac{3 x - 1}{4}$

multiply both sides by 5

$5 \left(x - 1\right) + 3 \left(2 x + 1\right) = 15 \frac{3 x - 1}{4}$

multiply both sides by 4

$20 \left(x - 1\right) + 12 \left(2 x + 1\right) = 15 \left(3 x - 1\right)$

$20 x - 20 + 24 x + 12 = 45 x - 15$

$20 x + 24 x - 45 x = - 15 + 20 - 12$

$- x = - 7$

multiply both sides by -1

$x = 7$

substitute x =7

$\frac{\left(7\right) - 1}{3} + \frac{2 \left(7\right) + 1}{5} = \frac{\left(3\right) \left(7\right) - 1}{4}$

$2 + 3 = \frac{20}{4}$

$5 = 5$