Assume toward a contradiction that x and y are integers satisfying 11x – 5y = 7,
+9x + 10y = –3. Then reducing both equations modulo 5 gives 1x – 0y ≡ 2 mod 5,
4x + 0y ≡ 2 mod 5. The first congruence is x ≡ 2 mod 5. Plugging this into the second congruence gives 3 ≡ 8 ≡ 4x ≡ 2 mod 5. But 3 and 2 are not congruent modulo 5, so this is a contradiction.