It is well--known that no general method exists for solving Diophantine equations. Thus, those Diophantine equations for which a complete, finite set of solutions can be found are often noteworthy. In this paper, using only elementary methods, we find all integer solutions a, b to the Diophantine equation 1/a + 1/b = (q+1)/pq, where p and q are distinct primes such that q + 1 divides p - 1. A restricted special case of this equation recently appeared in the 2018 William Lowell Putnam Mathematical Competition.