2.14. Cramer's Rule For Solving A System Of Three Linear Equations

A set of simultaneous equations

        a 1 x+ b 1 y+ c 1 z= d 1

        a 2 x+ b 2 y+ c 2 z= d 2                  (2.11)

        a 3 x+ b 3 y+ c 3 z= d 3

can be written as a single vector equation

        Ax+By+Cz=D                             (2.12)

in obvious notations.

Taking ( B×C )  on both sides gives

        A( B×C )x=D( B×C )

Provided A( B×C )0 , we have

                x= D( B×C ) A( B×C )   = | d 1 d 2 d 3 b 1 b 2 b 3 c 1 c 2 c 3 | | a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 |   = | d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 |

and analogously for y and z.  This is known as the Cramer's rule.

If A( B×C )=0 , then A, B, C lies in the same plane.  There will be no solution unless D also lies in the same plane.