Concerning decidability of a problem on real numbers
I think it's undecidable, but I don't have a proof. Intuitively, if $s\neq qr+p$ for all $p,q\in\mathbb{Q}$ (i.e. if the answer is "NO"), then you will be always looking for a "next" pair $(q,p)$ such that $|s-(qr+p)|$ is smaller than before. You don't really have an argument to decide when to stop and return "NO" because, for all you know, there is a better approximation to $s$ of the form $qr+p$ than the one you already have.