  In this work we investigate tensor completions of groups by associative
rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main
result states that there exists an algorithm that decides if a given finite
system of equations over a free ${\bf Q}$-group has a solution, and if it does,
finds a solution. This statement can be generalized for ${\bf Q}$-completions
of torsion-free hyperbolic groups. Our proof significantly uses the techniques
of word hyperbolic groups and the results of E.Rips and Z.Sela on the
solvability of systems of equations in hyperbolic groups.
