No integer solutions exist (6 is not divisible by $\gcd(42, 30)$)

Exactly one solution: $(x, y) = (1, -1)$

Infinitely many solutions of the form $x = -2 + 5t, y = 3 - 7t$ for integer $t$

Infinitely many solutions of the form $x = 2 - 5t, y = -3 + 7t$ for integer $t$