Cramer's Rule

x = Dx/D y = Dy/D z = Dz/D

Where D is the determinant of the coefficient matrix

Used to solve systems of equations using matrices and determinants

Ex. 1 3x - y + 2z = 4
2x + 3y - z = 14
7x - 4y + 3z = -4

x y z x y z
[ 3 -1 2 ] [ 4 -1 2 ]
D = [ 2 3 -1 ] = -30 Dx = [ 14 3 -1 ] = -30
[ 7 -4 3 ] [ -4 4 3 ]

x y z x y z
[ 3 4 2 ] [3 -1 4 ]
Dy = [ 2 14 -1 ] = -150 Dz = [ 2 3 14 ] = -90
[ 7 -4 3 ] [ 7 4 -4 ]

x = -30/-30 = 1 y = -150/-30 = 5 z = -90/-30 = 3

Ex. 2 7x + 4y = 19
3x - 10y = 14

D = [ 7 4 ] = -82 Dx = [ 19 4 ] = -246
[ 3 -10 ] [ 14 -10 ]

Dy = [ 7 19 ] = 41 x = -246/-82 = 3 y = 41/-82 = -1/2
[ 3 14 ]