Section 2.3
Systems of Linear Equations:
Underdetermined and
Overdetermined Systems
Theorem
1. If the number of equations is greater than or
equal to the number of variables then the
system has no solution, one solution, or
infinitely many solutions.
2. If the number of equations is less than or
equal to the number of variables, then the
system has no solution or infinitely many
solutions.
Ex. A system with no solution:
x  y  2z  3
3x  2 y  z  4
2x  3y  z  2
Matrix
1 1 2 3 


 3 2 1 4 
 2 3 1 2 
The system is inconsistent and
has NO solution.
 1 1 2 3


0 1 1 1
0 0 0 1
Notice the false
statement 0 = 1
Ex. A system with infinitely
many solutions:
x  y  2z  3
3x  2 y  z  4
2x  3y  z  1
Matrix
1 0 1 2 

 Notice
0 1 1 1  the row
0 0 0 0  of zeros.
1 1 2 3 


 3 2 1 4 
 2 3 1 1 
So
xz 2
y  z 1
or
x  2 z
y  1 z
If we let z = t then the solution is
given by
(2 – t, 1 – t, t)
Ex. A system with more equations
than variables:
x  3y  5
4 x  2 y  2
2x  3y  7
Matrix
1 3 5 


0
1
11/
7


0 0 4 / 7 
 1 3 5


 4 2 2 
 2 3 7 
Notice the false
statement
No Solution
Ex. A system with more
variables than equations:
x  y  3z  2w  1
3x  2 y  4 z  w  3
Matrix
1 1 3 2 1


3 2 4 1 3
So
x  2z  w  1
yzw0
1 0 2 1 1 


0 1 1 1 0
or
x  1 2z  w
y  z  w
Infinitely many solutions
If we let z = s and w = t then the solution is
given by
(1 – 2s + t, –s + t, s, t)