# How do you solve the system of linear equations 2x − 3y = 3 and 5x − 4y = 4?

Jul 8, 2017

There are two methods. 1. solve for one variable and then substitute.
2. Add or subtract the equations so that one variable is eliminated then substitute.

#### Explanation:

$2 x - 3 y = 3$ add 3y to both sides

$2 x - 3 y + 3 y = - 3 y + 3$ gives

$2 x = + 3 y + 3$ divide both sides by 2

$\frac{2 x}{2} = \frac{+ 3 y}{2} + \frac{3}{2}$ which gives.

$x = + \frac{3}{2} y + \frac{3}{2}$ Now substitute into the other equation

$5 x - 4 y = 4 = 5 \left(+ \frac{3}{2} y + \frac{3}{2}\right) - 4 y = 4$ which gives

$\left(+ \frac{15}{2}\right) y + \frac{15}{2} - 4 y = 4$ multiply everything by 2

$2 \left(+ \frac{15}{2}\right) y + 2 \left(\frac{15}{2}\right) - 4 y = 4$

$+ 15 y - 4 y + 15 = 4$ subtract 15 from both sides

$11 y + 15 - 15 = 4 - 15$

$11 y = - 11$ divide both sides by 11

 (11y/11 = -11/11 gives

$y = - 1$

$2 x - 3 \left(- 1\right) = 3$

$2 x + 3 = 3$ subtract 3 from both sides

$2 x + 3 - 3 = 3 - 3$ so

$2 x = 0$ divide both sides by two

$2 \frac{x}{2} = \frac{0}{2}$

$x = 0$