  Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow
$2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x
\in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By
combining this result with recent work of Breuer and Guralnick, we deduce that
if $G$ is a finite nonabelian simple group and $r$ is any prime divisor of
$|G|$, then $G$ is generated by a Sylow $2$-subgroup and a Sylow $r$-subgroup.
