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In this paper, we give evolution equations for free-choice Petri nets which generalize the [max, +]-algebraic setting already known for event graphs. These evolution equations can be seen as a coupling of two linear systems, a (min, +)-linear system and a quasi-(+, x)-linear one. This leads to new methods and algorithms to: 1) in the untimed case, check liveness and several other basic logical properties; 2) in the timed case, establish various conservation and monotonicity properties; and 3) in the stochastic case, check stability, i.e., the fact that the marking remains bounded in probability, and constructs minimal stationary regimes. The main tools for proving these properties are graph theory, idempotent algebras, and ergodic theory.