I have a system of non-linear equations:
$$x+y^3-2z=0$$
$$x+y=0$$
And I want to know if the solutions to this system is a subspace in $\mathbb R^3.$
I know that the solutions of a homogeneous linear system is a subspace in $\mathbb R^3.$ , but I am not sure if this still applies for a non-linear system. Could someone teach me how to determine whether the solutions to this system is a subspace or not?