GAUSS/JORDAN EXAMPLE "b"
FEWER EQUATIONS THAN VARIABLES
When there are fewer equations than variables, we proceed just as we did in
other examples of GAUSS/JORDAN. When the GAUSS/JORDAN
procedure has ended, however, we are typically faced with infinitely many
solutions. The student's dilemma is how to display such solutions. The problem
below has 3 variables and 2 equations; watch what happens below,
especially at the end:
Below is the system of equations which we will solve by G/J 


Below is the 1st augmented matrix :
pivot
on the"1" in the 11 position 

The one row operation for the first pivoting is
named below 
Next we pivot on the number
"1" in the 22 position, encircled in red below 



Below is the result of performing
P1 on the element "1"
in the 22 position. Next we must perform
P2 

The one row operation of
P2 is below 
The result of the second and last pivoting is below.
There is no 33 position, so the row operations of
GAUSS/JORDAN are complete 


Below is a copy of the final matrix of the previous line,
with it's identity matrix in blue 

We now rewrite each row of this matrix as an equation, just as
we wrote the original equations as a matrix 
A traditional style for a final solution
uses a parameter (here : z). Other variables are then written
in terms of that parameter (as below) 


The FINAL SOLUTION above contains infinitely
many points along the (RED) line AB below.
To get a sense for what this
solution "means" Prof M^{c}Farland has drawn graphs below
of the
original equations (blue and aqua planes below) and their
red intersection :