semigroups - Extend triple of coprime numbers to summation of coprime numbers

I apologize if this has already been answered or is common knowledge. If either of these is the case a reference will suffice.

Given $a, b, c$ positive integers which are pairwise relatively prime, do there exist positive integers $x,y,z$ such that
$$a x + b y = cz$$ and $ax, by, cz$ are pairwise relatively prime?