solution set of linear equation Is there a system of linear equations with 3 variables and solution set $\{(a, b, c) | a^2 =  b\}$?
My answer is no but I'm not sure how to explain it.
Edit: can't I just say that if $ax+by+cz=ax+a^2y+cz=d$ is the equation then also $2ax+4a^2y+2cz=2ax+2a^2y+2cz+2a^2y=\color{red}d$. but if I multiply $ax+a^2y+cz=d$ by 2 then I get that $2ax+2a^2y+2cz=2d$. then if I substitute this into the previous equation I get that $2d+2a^2y=d\Rightarrow d=-2a^2y$ which also means that $(0, -2a^2, 0)$ is solution, but it's true only for $a=0$.
Is this correct way to prove it?  

The solution of a system of linear equations is a vector subspace (the kernel of the mapping associated with the system) displaced by a vector (the particular solution).
If two variables in the solution have a quadratic dependence, any translation of the solution space will still have this dependence, since a translation just adds a constant to each vector component. Thus, the kernel of the linear map won't be a vector space, and won't be the solution of any homogeneous system.
