The set of vectors with $x + y + z = 0$ is a subspace, but $x + y + z = 1$ is not. Why does the second set fail to be a subspace?

C. It lacks the zero vector and is not closed under addition

A subspace must pass through the origin and stay closed. The second set is an affine plane — the same shape, moved — which is precisely why the solutions of $Ax = b$ are not a subspace while those of $Ax = 0$ are.