The set of all constant-free word equations satisfied by a given morphism is called an entire system of equations. We show that in the three-variable case, the set of nonperiodic solutions of any entire system can be described using parametric formulas with just one numerical parameter. We also show how the solution set of any equation can be represented as a union of solution sets of entire systems. Even though an infinite union is needed in some cases, this still points towards a stronger version of Hmelevskii’s theorem about parametric solutions of three-variable word equations.