Classically, in order to resolve an equation u ≈ v over a freemonoid X*, we reduce it by a suitable family $\cal F$ of substitutionsto a family of equations uf ≈ vf, $f\in\cal F$ , each involving lessvariables than u ≈ v, and then combine solutions of uf ≈ vfinto solutions of u ≈ v. The problem is to get $\cal F$ in a handyparametrized form. The method we propose consists in parametrizing thepath traces in the so called graph of prime equations associated tou ≈ v. We carry out such a parametrization in the case the primeequations in the graph involve at most three variables.