Undestandig of definition of an affine subspace

Question

Consider the affine space $\mathbb{R}^{3}$. Then for example the set $ \{(x,y,z):3x-3y+z=0\}$ is an affine subspace. But $ \{(x,y,z):3x-3y+z=2\} $ and $ \{(x,y,z):x^2\} $ are not affine subspaces. We have defined the affine subspace: $A = w_{0} + V = \{w_{0}+ v|v \in V \} $ and A is the affine subspace of $W$ ($W$ is a vector space and $V$ is vector subspace of $W$). And we defined $dim(A)=dim(V)$. Maybe it is quite simple, but I don't understand why $\{(x,y,z):3x-3y+z=2\}$ and $ \{(x,y,z):x^{2}\}$ is not an affine subspace and $\{(x,y,z):3x-3y+z=0\} $ is an subspace. Thanks in advance