Doubt on an identity vs equation intuition

Question

u+v+w=0 (1) -u+v+w=0 (2) Adding (1) and (2) we have 2v+2w=0 or, v=-w (3) Substituting (3) in (1) we have u=0 (4) y the place where they substituted v=-w in equation (1) while applying the Weiss Zone law is it permissible?I am asking this because I have used those two equations only to arrive at v=-w.Is this not an identity type of approach instead of an equation type?

Answer 1

Adding both equations we get $$v=-w$$, plugging this in one of the equations above we get $u-w+w=0$ so $u=0$ and we get $$[u,v,w]=[0,-w,w]$$