solving a generator system

[compsystems]

Hello, I have a question about solving a generator system.

augmented matrix A |X:

[[3,1,x1],
[2,1,x2],
[1,-2,x3]]


ref([[3,1,x1],[2,1,x2],[1,-2,x3]],keep_pivot) [enter]
[[1,-2,x3],
[0,5,x2-2*x3],
[0,0,5*x1-7*x2-x3]]

0=5*x1-7*x2-x3 (restriction in the last row )

last row
but

ref([[3,1,x1],[2,1,x2],[1,-2,x3]]) [enter]
[[1,-2,x3],
[0,1,(x2-2*x3)/5],
[0,0,1]]

last row
0=1

Why is there a solution in one case (with restrictions) and not in another (contradiction)?

[parisse]

keep_pivot prevents dividing by the pivot. Dividing does the implicit assumption that the pivot is not 0 (generically true in your example, but not always).