Example CFV Counting free variables

For each archetype that is a system of equations, the values of $n$ and $r$ are listed. Many also contain a few sample solutions. We can use this information profitably, as illustrated by four examples.

1. Archetype A has $n=3$ and $r=2$. It can be seen to be consistent by the sample solutions given. Its solution set then has $n-r=1$ free variables, and therefore will be infinite.
2. Archetype B has $n=3$ and $r=3$. It can be seen to be consistent by the single sample solution given. Its solution set can then be described with $n-r=0$ free variables, and therefore will have just the single solution.
3. Archetype H has $n=2$ and $r=3$. In this case, column 3 must be a pivot column, so by Theorem RCLS, the system is inconsistent. We should not try to apply Theorem FVCS to count free variables, since the theorem only applies to consistent systems. (What would happen if you did try to incorrectly apply Theorem FVCS?)
4. Archetype E has $n=4$ and $r=3$. However, by looking at the reduced row-echelon form of the augmented matrix, we find that column 5 is a pivot column. By Theorem RCLS we recognize the system as inconsistent.