Show that the system of equations Bx=b is consistent for all bεℝ³

B=[1 2 0 1;1 0 1 -1;0 -1 3 -4]

Please note that;means row; This is a 3 by3 matrix

Often in answers I find that they say let ( x y z ) be an element of R^3 and then they put it in the matrix(on the Rhs as a solution) and do reduced row echelon...Is that necessary here? why is that necessary in other cases?

(its possible that I dont need to do RREs but lets say i did ..is this method correct?)

Anyway heres what I have done.

I would appreciate feedback.THanks.

Please see attachments.