When we have less equations than unknowns , in general, there will be an
infinity of solutions (but still, not all values are , generally, solutions). Of
course, we still have the possibility of identities or contradictions, but there
will be no
" one solution " situation.
This is obvious in the case of two variables : in this case, if we have
less equations than 2, we have one. One equation
defines a line in the plane , and that will be the solution set .
The case of three (or more) unknowns is more interesting .
Geometrically, a linear equation in three variables defines a plane. Thus, if we
have one equation, that plane is the solution set. If we have two equations, we
have the following
possibilities:
•The corresponding planes are parallel: no solution
•The corresponding planes coincide: they are the solution set
•The corresponding planes do neither: in this case, they will meet in a line,
and that line will be the solution set.
All of this can be de termined algebraically , without
worrying about pictures - which is good, since, if we move on, to four or more
unknowns, there are no pictures available at all!
As an example, consider the fol lowing system
Note that we cannot use Cramer's rule to handle this
problem: the matrix of the coefficients is not square , so we cannot
define a \determinant" (there are ways to use a determinant-related approach to
a problem like this , but that's
way beyond our scope).
We can solve this problem through one of the other
two methods we know : substitution or elimination . As an example, use
elimination , adding the two equations . This yields
4x - y = 0
If we substitute y = 4x in, say, the first equation we get
5x + z = 1
We can also substitute (or eliminate) in other ways, but
we never get down to a form like 
we always
find ourselves with two equations. All these pairs are equivalent : they define
pairs of planes, all intersecting on the
same line, which is the solution set.
