Question

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

Answer 1

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`