# How do you solve the system using the elimination method for 4x-5y=27 and 3x-9y=36?

Jul 2, 2015

The solution of this system is: $\left\{\left(3 , - 3\right)\right\}$

#### Explanation:

To use the elimination method, you firts have to choose which variable you're going to eliminate. I will eliminate the $y$ variable.

When you have a system of equations there are several transformations which are allowed on the system:

1. Multiply one of the equations completely (by the same number on both sides).
2. Add equations of the system.

By combining the two, you can easily eliminate the $y$ variable:

$4 x - 5 y = 27$
$3 x - 9 y = 36$

$9 \cdot \left(4 x - 5 y\right) = 9 \cdot 27$
$- 5 \cdot \left(3 x - 9 y\right) = - 5 \cdot 36$

How did I know by which number to multiply? Well, you take the LCM of the coefficients of the variable you want to eliminate. Then you also need to make sure that the resulting coefficients of the variable have opposite signs, sou if you add them they will be eliminated.

$36 x - 45 y = 243$
$- 15 x + 45 y = - 180$

$36 x - 45 y - 15 x + 45 y = 243 - 180$
$21 x = 63$
$x = 3$

At this point, you can repeat the process and eliminate $x$ to find the value of $y$, but what I like to do, is just fill in the value of $x$ that you've just found in one of the equations:

$4 \cdot 3 - 5 y = 27$
$12 - 5 y = 27$
$- 5 y = 15$
$y = - 3$

So the solution of this system is: $\left\{\left(3 , - 3\right)\right\}$