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
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
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.