Help in diophantine equations

I have these diophantine equations:

  1. x(m + 2y) = n
  2. x(m + 3y) = n

where m and n are some integers. How do I check if solutions exist for these equations (where x and y are integers)?

  1. I don’t need the solutions. I just need to determine if solutions exist.
  2. This is NOT a system of equations. These are unique equations.

I thought of using Bezout’s lemma but I couldn’t apply it.