How to solve system of congruence modulo equations?

Question

So I have these two congruence equations, and I need to solve for x and y (or determine that it is unsolvable). I have been searching for hours and can't figure out how to solve this, or how to even find out if it is solvable.

Here are the two equations:

1. x + 3 y ≡ 3 (mod 6)
2. 4 x + 3 y ≡ 1 (mod 6)

Thanks in advance

Answer 1

If you subtract (1) from (2) the system implies $3x \equiv -2 \equiv 4 \ (\mod 6)$ but $3x$ can only be $0$ or $3$ mod $6.$