Graphics Reference
In-Depth Information
the value of
x
is defined as
d
1
b
2
c
3
−
d
1
b
3
c
2
+
d
2
b
3
c
1
−
d
2
b
1
c
3
+
d
3
b
1
c
2
−
d
3
b
2
c
1
x
=
(7.112)
a
1
b
2
c
3
−
a
1
b
3
c
2
+
a
2
b
3
c
1
−
a
2
b
1
c
3
+
a
3
b
1
c
2
−
a
3
b
2
c
1
with similar expressions for
y
and
z
. Once more, the denominator comes from
the determinant of the matrix associated with the matrix formulation of the
linear equations:
⎡
⎤
⎡
⎤
⎡
⎤
d
1
d
2
d
3
a
1
b
1
c
1
x
y
z
⎣
⎦
=
⎣
⎦
·
⎣
⎦
a
2
b
2
c
2
(7.113)
a
3
b
3
c
3
where
a
1
b
1
c
1
a
2
b
2
c
2
=
a
1
b
2
c
3
−
a
1
b
3
c
2
+
a
2
b
3
c
1
−
a
2
b
1
c
3
+
a
3
b
1
c
2
−
a
3
b
2
c
1
a
3
b
3
c
3
which can be written as
a
1
−
a
2
+
a
3
b
2
c
2
b
1
c
1
b
1
c
1
(7.114)
b
3
c
3
b
3
c
3
b
2
c
2
Let's now see what creates a zero determinant. If we write, for example
10 = 2
x
+
y
(7.115)
there are an infinite number of solutions for
x
and
y
, and it is impossible
to solve the equation. However, if we introduce a second equation relating
x
and
y
:
4=5
x
−
y
(7.116)
we can solve for
x
and
y
using (7.107):
x
=
10
×
(
−
1)
−
4
×
1
2
=
−
14
−
=2
×
(
−
1)
−
5
×
1
7
2
×
4
−
5
×
10
1
=
−
42
−
y
=
= 6
(7.117)
2
×
(
−
1)
−
5
×
7
therefore
x
=2and
y
= 6, which is correct.
But say the second equation had been
20 = 4
x
+2
y
(7.118)
which would have created the pair of simultaneous equations
10 = 2
x
+
y
(7.119)
20 = 4
x
+2
y
(7.120)
