Summary Solving systems of equations using Cramer's Rule

Solve the following system of equations using Cramer’s Rule


5x +2y −z = −7
x −2y +2z = 0
3y +z = 17



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|

 
−7
We have B =  0 
17

Coefficient matrix
 
5 2 −1
D = 1 −2 2 
0 3 1

We calculate the determinants
5 2 −1
|D| = 1 −2 2 = −10 + 0 + (−3) − (0 + 2 + 30) = −45
0 3 1

Similarly,
−7 2 −1
|Dx | = 0 −2 2 = 14 + 68 + 0 − (34 + 0 + (−42)) = 90
17 3 1

5 −7 −1
|Dy | = 1 0 2 = 0 + 0 + (−17) − (0 + (−7) + 170) = −180
0 17 1

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