# Solving linear systems using Cramer's Rule

### Solving linear systems using Cramer's Rule

Last chapter we saw that we are able to solve linear systems with Gaussian Elimination. Now we are going to take a look at a new method which involves solving linear systems with Cramer's Rule. Cramer's Rule requires us to find the determinant of 2 x 2 and 3 x 3 matrices (depends on your linear system). However, this rule can only be used if you have the same number of equations and variables. If you have a different number of equations and variables, then finding the determinant will be impossible. Hence, it will not be possible to use Cramer's rule.

#### Lessons

This is a different way of solving linear systems. Instead of using Gaussian Eliminations, you can use Cramer's Rule! Make sure to review your determinants of 2 x 2 and 3 x 3 matrices.

Cramer's Rule for 2 x 2 matrices:
$x=\frac{D_x}{D}$

$y=\frac{D_y}{D}$

Cramer's Rule for 3 x 3 matrices:
$x=\frac{D_x}{D}$

$y=\frac{D_y}{D}$

$z=\frac{D_z}{D}$

• Introduction
Cramer's Rule Overview:
a)
Using Cramer's Rule with 2 x 2 matrices

b)
Using Cramer's Rule with 3 x 3 matrices

• 1.
Cramer's Rule with 2 x 2 matrices
Solve the following linear systems with Cramer's Rule"
a)
$x+2y=3$
$2x+3y=1$

b)
$5x+3y=1$
$x+y=2$

c)
$y=3x+5$
$y=4x-2$

d)
$2x+4y=3$
$4x+8y=6$

• 2.
Cramer's Rule with 3 x 3 matrices
Solve the following linear systems with Cramer's Rule"
a)
$x+4y+3z=1$
$x+2y+9z=1$
$x+6y+6z=1$

b)
$x+3y+4z=4$
$-x+3y+2z=2$
$3x+9y+6z=-6$

c)
$2-3y-3z=x$
$3x+9y=3-3z$
$3x+6y+6z-4=0$