How do I use Cramer's rule to solve a system of equations?

1 Answer
Aug 16, 2015

You divide each variable determinant by the determinant of the coefficients.

EXAMPLE:

Use Cramer's Rule to solve the following system of equations:

2x+y+z=3
x–y–z=0
x+2y+z=0

Solution:

The left hand side gives us the coefficient matrix.

((2,1,1),(1,-1,-1),(1,2,1))

The right hand side gives us the answer matrix.

((3),(0),(0))

The determinant D of the coefficient matrix is

D = |(2,1,1),(1,-1,-1),(1,2,1)| = -2+2-1+1-1+4=3

Let D_x be the determinant formed by replacing the x-column values with the answer-column values:

D_x=|(3,1,1),(0,-1,-1),(0,2,1)|= -3+6=3

Similarly,

D_y=|(2,3,1),(1,0,-1),(1,0,1)|= -3-3=-6

and

D_z=|(2,1,3),(1,-1,0),(1,2,0)| = 6+3=9

Cramer's Rule says that

x = D_x/D =3/3=1,

y = D_y/D=-6/3=-2,

z = D_z/D=9/3=3.

The solution is x=1,y=-2,z=3