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

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Answer 1 · 2015-08-16T16:27:45

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#