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.