examples of subspaces?

Question

is this equation an example of a subspace x+2y-3z=1

Answer 1

No. Most simply, the origin is not in this subset, so it can't be a subspace.

Note that it is not closed under addition (add any vector in this subset to itself and the result won't be in the subset), nor under scalar multiplication (if x + 2y - 3z = 1, then (kx) + 2(ky) - 3(kz) = k, so multiplying by any scalar other than 1 will take you outside the set).