Systems of Linear Equations with Infinitely Many
Solutions
When does a system of linear equations have infinitely
many solutions? This will happen when
-
There is at least one solution to the system and there are more unknowns
than equations, or
-
There is at least one solution to the system and the system can be transformed
to an equivalent one with more unknowns than equations.
Example Consider the one-equation system
There are many solutions to this equation; for example (x,
y) = (4, 0) is a solution. Moreover, for any value we
pick for one of the variaibles, there is a corresoponding value for the
other one so that together they are a solution. Try
for yourself and see that this works! In fact, all the points
on the line defined by this equation are solutions of the system.
Example Consider the system of 3
equations in 3 unknowns
x +
y - z = 1
2x - y
= 0
3y - 2z = 2 |
This system is equivalent
to the system
x + y
- z = 1
-3y + 2z
= -2 |
which has more equations than variables. Since the systems are equivalent,
they have the same solution sets.
Geometric Interpretation Each of
the equations describes a plane in 3-dimensional space, and the solutions
are the straight line where these planes intercept each other.
How do we find solutions in this kind of situation?
-
The first step has alreaady been taken, by transforming the original system
of equations to the second, in which there are no extraneous equations.
-
The next step is to give one of the variables an arbitrary value; say z
= t and to use that to write values for the other variables
using this arbitray value. From the second equation we see that
-3y
= -2 -2z = -2 - 2t
-3y = -2(1 + t)
y = 2(1 + t)/3 |
and from the first equation we now get
x = 1 -
y + z
x = 1 - 2(1 + t)/3 +
t |
Thus the solutions are all triples of the form (x,
y, z) = ( 1 - 2(1 + t)/3 + t,
2(1 + t)/3, t ).
This kind of solution is called a parametric solution,
since it depends on the parameter t.