Solve the system of congruences

Question

Solve the system of congruences

( 2x + 6y ≡ 3 (mod 15)

3x + 7y ≡ 6 (mod 15)

Answer 1

Subtract the first congruence from the second, and we obtain x + y ≡ 3(mod 15). Multiply this by 2, and subtract it from the first congruence, and we have 4y ≡ -3 ≡ 12(mod 15). Since 4 and 15 are relatively prime, we can divide by 4 to get y ≡ 3(mod 15). Substituting this into x + y ≡ 3(mod 15), we have x ≡ 0(mod 15). The solution (0,3) fits both original congruences.