Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\geq 0$, let $0\leq r\leq n$, and set $R:=k[x_1,\dots,x_n]$. Let $I\trianglelefteq R$ be an ideal and set $X:=V(I)\subseteq \mathbb A_k^n$. Let $S:=k[x_{r+1},\dots,x_n]\subseteq R$, with the convention that $S=k$ when $r=n$, and define the $r$-th elimination ideal by $I_r:=I\cap S$. Let $\pi:\mathbb A_k^n\to \mathbb A_k^{n-r}$ be the coordinate projection sending $(a_1,\dots,a_n)$ to $(a_{r+1},\dots,a_n)$. Then $\overline{\pi(X)}=V(I_r)\subseteq \mathbb A_k^{n-r}$, where the closure is taken in the Zariski topology on $\mathbb A_k^{n-r}$.