First we evaluate the matrix corresponding to the coefficients of the variables.
axx+ayy+azz=A,
bxx+byy+bzz=B,
cxx+cyy+czz=C,
represent the system of equations with the coefficients and the constants. Cramer's Rule can be used. First evaluate:
⎟ax ay az⎟
⎟bx by bz⎟
⎟cx cy cz⎟
Call this determinant D. If D=0, the system is inconsistent or insufficient.
Then we evaluate three determinants, one for each variable:
Dx=
⎟A ay az⎟
⎟B by bz⎟
⎟C cy cz⎟
From this x=Dx/D.
Dx=
⎟A ay az⎟
⎟B by bz⎟
⎟C cy cz⎟
From this x=Dx/D.
Dy=
⎟ax A az⎟
⎟bx B bz⎟
⎟cx C cz⎟
From this y=Dy/D.
Dz=
⎟ax ay A⎟
⎟bx by B⎟
⎟cx cy C⎟
From this z=Dz/D.