Explanation

5x + 9y = 7 gives x = (7 − 9y)/5, so y ≡ 3 (mod 5); write y = 5k + 3, then x = −4 − 9k. The bound −500 < x < 500 gives −55 ≤ k ≤ 55, and the corresponding y values also lie within (−500, 500). That is 111 values of k, hence 111 pairs.