  Hyperfields and systems are two algebraic frameworks which have been
developed to provide a unified approach to classical and tropical structures.
All hyperfields, and more generally hyperrings, can be represented by systems.
Conversely, we show that the systems arising in this way, called {\it
hypersystems}, are characterized by certain elimination axioms. Systems are
preserved under standard algebraic constructions; for instance matrices and
polynomials over hypersystems are systems, but not hypersystems. We illustrate
these results by discussing several examples of systems and hyperfields, and
constructions like matroids over systems.
