To show that the set of all solutions of the system $ax+by+cz =0, dx+ey+cz=0$ forms a subspace.
Subtracting we get $x = \frac{e-b}{a-d}y$ and putting this in any one of the equations we get $z= \frac{bd-ae}{c(a-d)}y$.
Thus the solution set is span$\{(e-b,a-d,\frac{bd-ae}{c} )^T\}$ and hence a subspace.
Is the solution correct?