Graphics Reference
In-Depth Information
Multiplying the first equation by
d
and the second by
b
gives
adx
+
bdy
=
de
, and
bcx
+
bdy
=
bf
.
Subtracting the second equation from first gives
adx
−
bcx
=
de
−
bf
, from which
x
can
be solved for as
x
bc
).
The solution to this system corresponds to finding the intersection point of two
straight lines, and thus three types of solution are possible: the lines intersect in a
single point, the lines are parallel and nonintersecting, or the lines are parallel and
coinciding. A unique solution exists only in the first case, signified by the denominator
ad
=
(
de
−
bf
)/(
ad
−
bc
). A similar process gives
y
=
(
af
−
ce
)/(
ad
−
−
bc
being nonzero. Note that this 2
×
2 system can also be written as the matrix
equation
AX
=
B
, where
ab
cd
,
X
x
y
,
e
f
.
A
=
=
and
B
=
A
is called the
coefficient matrix
,
X
the
solution vector
, and
B
the
constant vector
.A
system of linear equations has a unique solution if and only if the determinant of the
coefficient matrix is nonzero,
A
−
1
B
.
Upon examining the previous solution in
x
and
y
it becomes clear it can be
expressed in terms of ratios of determinants:
|
A
| =
0. In this case, the solution is given by
X
=
eb
f d
ae
cf
x
=
,
y
=
.
ab
cd
ab
cd
Here, the denominator is the determinant of the coefficient matrix. The
x
numerator
is the determinant of the coefficient matrix where the first column has been replaced
by the constant vector. Similarly, the
y
numerator is the determinant of the coefficient
matrix where the second column has been replaced by the constant vector.
Called
Cramer's rule
, this procedure extends to larger systems in the same manner,
allowing a given variable to be computed by dividing the determinant of the coef-
ficient matrix (where the variable column is replaced by the constant vector) by the
determinant of the original coefficient matrix. For example, for the 3
×
3 system
a
1
x
+
b
1
y
+
c
1
z
=
d
1
,
a
2
x
+
b
2
y
+
c
2
z
=
d
2
, and
a
3
x
+
b
3
y
+
c
3
z
=
d
3
