7.3 Independent, Dependent and Inconsistent Systems


An independent system of equations is a system with a unique solution. All the examples and exercises in 7.2 were examples of independent systems.

Consider the following system:

S1
| x + y + z = 1
|     y + z = 2

Since there are fewer equations in the system than variables, there will not be a unique solution. In order to characterize the solutions of this system, we add a third equation to represent all possible values of z:
S2
| x + y + z = 1
|     y + z = 2
|         z = t

where t can be any real number. Using back substitution, we find
S3
| x + y + z = 1
|     y     = 2 - t
|         z = t

and
S4


So the solution is {( - 1, 2 - t, t ) | t is a 
real number }.  This is an example of a dependent system..  
Dependent systems have an infinite number of solutions.



Exercise 7.3.1



Solve the system


S1
| x + 2 y  + z = 0
|       y  - z = 2




Solution



Consider the system


S1
| x + y + z = 1
|     y + z = 2<
|     y + z = 3




If we attempt to triangularize the system by E3 -> - E2 + E3 we get the 
system


S2
| x + y + z = 1
|     y + z = 2
|         0 = 1




Since the last equation is false, this system has no solution.  The system 
is said to be inconsistent.



Exercise 7.3.2



Solve the system


S1
| x  - y + z = 1
| x      + z = 1
| x  + y + z = 2




Solution



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