Game Development Reference
In-Depth Information
An Infinite Number of Solutions
If an inconsistent system has no solutions, another type of system, known as a
dependent system, possesses an infinite number of solutions. The reason this
occurs is that when you evaluate such systems, you find that you can express one
variable in terms of the other. You have at hand such a system when you can
multiply one of the equations in the system by some value that produces an
equation that is the same as the other equation in the system. Equations that
possess such a relationship with each other are known as dependent equations.
To see how this can happen, consider this system of equations:
4
x þ
6
y ¼
2
8
x þ
12
y ¼
4
To make it easier to work with the two equations, reverse them:
8
x þ
12
y ¼
4
4
x þ
6
y ¼
2
Then to make it so that you can eliminate one of the variables, multiply the
second equation by
2 to create an equivalent equation:
4
x þ
6
y ¼
2
ð
multiply by
2
Þ
The outcome of this activity is this equation:
8
x
12
y ¼
4
If you then add this equation to the first of the equations in the system, your
activity proceeds along the following lines:
8x
þ
12y
¼
4
8
x
12
y ¼
4
0
þ
0
¼
0
The system you are dealing with, then, consists of one equation expressed in two
different ways, so to reach a solution for the system, you can solve 8
x þ
12
y ¼
4
for
x
and
y
. One approach to this involves substitution and finding the solution
