According to the section Definition and expression in homogeneous coordinates of this Wikipedia article, we get
\begin{align} y_1 &= \frac{a_{1,0} + a_{1,1}x_1 +\dots + a_{1,n}x_n}{a_{0,0} + a_{0,1}x_1 +\dots + a_{0,n}x_n}\\ &\vdots\\ y_n &= \frac{a_{n,0} + a_{n,1}x_1 + \dots + a_{n,n}x_n}{a_{0,0} + a_{0,1}x_1 +\dots + a_{0,n}x_n} \end{align}
which is induced by a definition of projective space (Affine Spaces + points at infinity).
I'm wondering how we lost $x_0, y_0$ from
\begin{align} y_0 &= a_{0,0}x_0 +\dots + a_{0,n}x_n\\ &\vdots\\ y_n &= a_{n,0}x_0 +\dots + a_{n,n}x_n. \end{align}
Is it because we are assuming $x_0 \neq 0$, $y_0 \neq 0$? In this way, $y_1,...,y_n$ can be devided by $y_0$, and $x_1, ..., x_n$ can be devided by $x_0$.
But, then why the first coordinates don't look like
$$\frac{y_n}{y_0}=\frac{a_{n,0}x_0 +\dots + a_{n,n}x_n}{a_{0,0}x_0 +\dots + a_{0,n}x_n} \\ =\frac{a_{n,0} +\dots + a_{n,n}\frac{x_n}{x_0}}{a_{0,0} +\dots + a_{0,n}\frac{x_n}{x_0}}$$??
I'm thinking $\frac{y_n}{y_0}$ is relabeled with $y_n$ and $\frac{x_i}{x_0}$ for each $i=1,...,n$ is relabled with $x_i$.
Is it correct?