In science, matter-energy is an oft-used, albeit ill-defined term, referring to a matter-converting-into-energy type of quantity, according to the mass-energy equivalency relation, E = mc², such as occurs in a nuclear reaction. The following illogical 1978 quote by American psychologist James Miller gives one example of its use: [1]

“Most concrete systems have boundaries which are at least partially permeable, permitting sizable magnitudes of at least certain sorts of matter-energy or information transmissions to cross them … such a system is an open system.”

The term “matter-energy” can also allude to the logic of a piece of matter, such as coal, producing energy, such as heat, through a combustion reaction. The following 1989 quote by American economist Jeremy Rifkin gives an example of the term used in this manner: [2]

“[a point in time in which] the particular matter-energy base that a society is using becomes depleted, as a result of natural forces at work or as a result of people consuming resources faster than nature can reproduce them.”

Some publications even use the phrase the “law of conservation of matter/energy”; thus negating a prolonged discussion on the actuality of two separate laws: the law of conservation of energy, and the conservation of mass.

In his 1990 social entropy theory, American sociologist Kenneth Bailey mixes together the idea of “matter-energy”, with Ludwig Bertalanffy’s general systems theory, and mis-defined thermodynamics terms, to state that “since Einstein, matter-energy can be hyphenated, and only one type of closed system need be designated: [one] closed against transfers of matter-energy across its boundaries.” [3] In the correct thermodynamic sense, a system can have only one boundary and "closed" means that energy can pass, but not matter.

Difficulties on term
The term “matter-energy” tends to be used a metaphorical type of quantity by those who are looking only for generalizations. The rigor of the term looses value when the details of any particular transformation are broken down thermodynamically or via particle physics.

References
1. Miller, James G. (1978). Living Systems, (pgs. 11, 17-18). McGraw-Hill.
2. Rifkin, Jeremy. (1989). Entropy: Into the Greenhouse World (revised edition) (pgs. 293-94). New York: Bantam.
3. Bailey, Kenneth D. (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis (pg. 151). SUNY Press.