Here is a rewriting of the question, using only the explanations finally provided by the OP.
Assume that $A\cdot x=b$ and $A\cdot y=0$. Show that, for every $t$ in $\mathbb C$, $A\cdot(x+ty)=b$.
What this formulation shows, I believe, is the following:
- All the LS, N, w stuff in the original post is not needed and may be more an obstacle than a help to the comprehension of the question.
- Crucial hypotheses are implicit, which are that one is working with some vector spaces $V$ and $W$ (over $\mathbb C$), that $x$ and $y$ belong to $V$ and $b$ belongs to $W$, and that $A:V\to W$ is linear.
- The proof is direct once the definitions related to these hypotheses are recalled.
For example, to reduce the general case to the $t=1$ case, one could first show that $A\cdot(ty)=0$. Can you do that?