Summary Solving systems of equations using Cramer's Rule

Solve the following system of equations using Cramer’s Rule


 4x −y +3z = 6
−8x +3y −5z = −6
5x −4y = −9



Explanation and answer.

Cramer’s Rule

For the given system. Let D the coefficient matrix. Let Dx the matrix obtained replacing
the x-coefficient by B-coefficient. Let Dy the matrix obtained replacing the y-coefficient by
B-coefficient. Let Dz the matrix obtained replacing the z-coefficient by B-coefficient.

When the determinant |D| =
6 0 the solutions are given by

|Dx | |Dy | |Dz |
x= ,y= ,z=
|D| |D| |D|



6
We have B = −6
−9

Coefficient matrix
 
4 −1 3
D = −8 3 −5
5 −4 0

We calculate the determinants
4 −1 3
|D| = −8 3 −5 = 0 + 25 + 96 − (45 + 0 + 80) = −4
5 −4 0

Similarly,
6 −1 3
|Dx | = −6 3 −5 = 0 + (−45) + 72 − (−81 + 0 + 120) = −12
−9 −4 0

4 6 3
|Dy | = −8 −6 −5 = 0 + (−150) + 216 − (−90 + 0 + 180) = −24
5 −9 0

1