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In spectroscopy, the isosbestic point is said to be where the molar absorptivities of two species is the same.
$$A= \epsilon(X+Y) \times c(X+Y) \times l$$
reduces to $$A= \epsilon_t \times c \times l$$
the absorbances and the length are the same at this point, but the concentrations are not;
why are the molar absorptivities defined to be the same?
Is c taken to be the relative contribution, rather than the exact concentration?
An isosbestic point is found by maintaining the sum of the concentrations of the two (or more, though it's very unlikely) species the same and varying the individual contributions of each, and looking for the point at which the absorbance spectra intersect. This is easy for something like a pH indicator where a pH change just converts one species into the other—the sum of the two species is always the same. Under these conditions, a point where the absorbance doesn't change with the ratio of the two species can only exist if the molar absorptivities of the two species are the same, otherwise changing their ratio would change the absorbance.
So the molar absorptivities are defined to be the same when you simplify your equation above because this is the only way an isosbestic point can exist. So in your simplified equation, c is the sum of the concentrations of both species as that is what determines the net absorbance, not the concentrations of each individual species.
Actually that's not the correct definition of the isosbestic point. From IUPAC goldbook:
Isosbestic point - Wavelength, wavenumber or frequency at which the total absorbance of a sample does not change during a chemical reaction or a physical change of the sample.
A simple example occurs when one molecular entity is converted into another that has the same molar absorption coefficient at a given wavelength. As long as the sum of the concentrations of the two molecular entities in the solution is held constant there will be no change in absorbance at this wavelength as the ratio of the concentrations of the two entities is varied.
Contrary to a widely accepted idea, the existence of an isosbestic point does not prove that the reaction is a quantitative conversion of one species into a unique other species or that an equilibrium exists between only two species. The observation of isosbestic points only indicates that the stoichiometry of the reaction remains unchanged during the chemical reaction or the physical change of the sample, and that no secondary reactions occur during the considered time range, For the reaction A + B → c C + d D + e E, with c, d, and e the percentages of the products C, D, and E, an isosbestic point will be observed at every wavelength where the condition ɛ A + ɛ B = c ɛ C + d ɛ D + e ɛ E, provided that the values of the percentages c, d, and e remain constant during the chemical reaction or the physical change. The use of the obsolete term isoabsorption point is not recommended.
