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.">

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=DxDx=\frac{D_x}{D}

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

Cramer’s Rule for 3 x 3 matrices:
x=DxDx=\frac{D_x}{D}

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

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

  • 1.
    Cramer’s Rule Overview:
    a)
    Using Cramer’s Rule with 2 x 2 matrices

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


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

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

    c)
    y=3x+5 y=3x+5
    y=4x2 y=4x-2

    d)
    2x+4y=3 2x+4y=3
    4x+8y=64x+8y=6


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

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

    c)
    23y3z=x 2-3y-3z=x
    3x+9y=33z 3x+9y=3-3z
    3x+6y+6z4=0 3x+6y+6z-4=0