We decided that if (a,b,c) is a solution, then b=-a, c=a. So the solution really looks like (a,-a,a) for any a. We can factor out the a to get a*(1,-1,1) which is just the set of linear combinations of the basis of a straight line where the basis is (1,-1,1)when are saying (x*(1,-1,-1) is a solution for any x.) what do mean by that ?
True moo, it needs to have a vector in it also
Neonn, you're on the right track in your first post.
Like you said, x+y=w solves the case of s=0. (note z can be whatever you want
So our solution looks like (x,y,z,x+y)
Or we can make that basis by separating it
x*(1,0,0,1) + y*(0,1,0,1) + z*(0,0,1,0) (you should see or be able to prove that's a basis) and you can see summming those vectors gives you a solution to the original equation for any x,y,z.
Do the same for s=2