Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are prime numbers? How about a special case when $b=d=1$?

This seems like something that should hold, some kind of a generalization of Dirichlet's theorem, but I wasn't able to show it.