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In 1905, Einstein carried out his first derivation of the mass-energy equivalence by studying in different reference frames the energy balance of a body emitting electromagnetic radiation and assuming special relativity as a prerequisite. In this paper, we prove that a general mass-energy relationship can be derived solely from very basic assumptions, which are the same made in Einstein's first derivation but completely neglecting special relativity. The general mass-energy relationship turns to a mass-energy equivalence when is applied to the case of a body emitting energy in the form of electromagnetic waves. Our main result is that if the core logic behind Einstein's approach is sound, then the essence of the mass-energy equivalence can be derived without special relativity. We believe that our heuristic approach, although not capable of giving the exact mathematical formula for the mass-energy equivalence, may represent a useful addition to the general discussion on the matter at the graduate level. Our finding suggests that the connection between mass and energy is at a deeper level and comes before any full-fledged physical theory.