Can any system be solved using the addition and subtraction method?
Yes, any LINEAR systems with sufficient equations (one per variable) can be solved in this way.
Basically, that is what solving linear "systems" of equations is all about. Just as multiplication and division depend on addition and subtraction in algebra, "advanced" methods for solving linear equation systems are still ultimately based on the addition/subtraction method of variable reduction.
This does NOT apply universally to ANY system of equations.