We study systems of equations Ni(t)D(t) = ki (i = 1, . . . , n) that have a common solution t0, possibly together with equations Fi(x) = ki in a second unknown x with the same right-hand sides ki. Adding or subtracting these equations produces new systems whose solutions other than t0 (the hidden solutions) are computed by the rule a b Θ c d = a+c b+d (the Farey mediant), provided the orientation −A/B ̸= A/(−B) is respected. A second law, denoted ♢, says how to multiply an equation by a coecient: λ♢a b = λa λb , which is not the usual productof a fraction by a number. We explain these rules by a factorisation theorem, determinethe signs to choose in order to obtain a prescribed hidden solution, and give an inverseconstruction, relations between equations, a barycentric formula in degree m, the complexcase in degree 2, covariance under homographies, the correspondence between the unknownsx and t, two-parameter systems (relations between hidden values in t and in α, elliptic curves of rank ≥ 1), and a study of the rationality of all solutions (conics, elliptic curves). A rank criterion proved with Mazur's theorem gives innitely many fully rational combinations for certain pencils of cubics. Every statement is proved and illustrated by explicit systems. The core is elementary (Vieta's relations and linearity); the contribution is the reading of thesesystems and of their signed combinations.