Did you actually mean to ask how to prove that the set of all solutions in $\,\Bbb R^3\,$ (or in any other vector space) to the given linear system of equations is a subspace? Well, prove that the sum of any two of them is again a solution, and the product of any of them by any scalar is also a solution...oh, and to avoid the case of an empty set, prove the zero vector is a solution, too. – 2012-12-18