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If free variables are the variables that aren't basic variables, then how did this example come up with x4?

Reduced echelon form:

-1 0 2 0 
0 -1 3 0 
0  0 0 0  
0  0 0 0  

x1 = 2x3
x2 = 3x3
x3 and x4 are free
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up vote 3 down vote accepted

Basic variables corresponding to columns containing an entry that is the first nonzero entry in a row: in this case $x_1$ and $x_2$. Anything else is free. That includes cases where the column has no nonzero entries at all, which is what you have with $x_4$ in your example.

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