# How do I use substitution to find the solution of the system of equations y=1/3x+7/3 and y=-5/4x+11/4?

Sep 26, 2014

In this problem we have 2 different expressions that represent the value of $y$. Because they both represent $y$ we can set them equal to each other.

$\left(\frac{1}{3}\right) x + \frac{7}{3} = - \left(\frac{5}{4}\right) x + \frac{11}{4}$

Because all of the terms share a common multiple of 12 I will multiply each term by 12 to eliminate the fractions.

$12 \left[\left(\frac{1}{3}\right) x + \frac{7}{3} = - \left(\frac{5}{4}\right) x + \frac{11}{4}\right]$

$\left[\left(\frac{12}{3}\right) x + \frac{84}{3} = - \left(\frac{60}{4}\right) x + \frac{132}{4}\right]$

$4 x + 28 = - 15 x + 33$

$19 x = 5$

$x = \frac{5}{19}$

Substitute in this newly found $x$-value into one of the original equations.

$y = \left(\frac{1}{3}\right) x + \frac{7}{3}$

$y = \left(\frac{1}{3}\right) \cdot \left(\frac{5}{19}\right) + \frac{7}{3}$

$y = \left(\frac{5}{57}\right) + \frac{7}{3} \cdot \frac{19}{19}$

$y = \frac{5}{57} + \frac{133}{57}$

$y = \frac{138}{57}$

$y = \frac{46}{19}$