On page 221 of Vakil's FOAG, he states The Fundamental Theorem of Elimination theroy as follows:
The morphism $\pi : \mathbb{P}_A^n \to \mathrm{Spec}A $ is closed.(sends closed sets to closed sets)
Then he gives some examples as special cases of this theorem. The first example is as follows:
First, let $A=k[a,b,c,...,i]$, and consider the closed subset of $\mathbb{P}_A^2$(taken with coordinates $x,y,z$)corresponding to $ax+by+cz=0, dx+ey+fz=0,gx+hy+iz=0$. Then we are looking for the locus in $\mathrm{Spec}A$ where these equations have a nontrival solution. This indeed corresponds to zariski-closed set---where $\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=0$.
Firstly, I can't figure out why this is a special case of the fundamental theorem. Maybe the image of $\pi$ in $\mathrm{Spec}A$ is where these equations have a nontrival solution? But I can't see how to prove it. And, if it is the case, in what sense a nonclosed point of $\mathrm{Spec}A$ could become coefficient of equations? (if it's of the form $(a-a_0,...,i-i_0)$, it would make sense, but what if it's not?)