  In this paper we study the difference between algebraic and geometric
solutions of the hyperbolic Dehn filling equations for ideally triangulated
3-manifolds. We show that any geometric solution is an algebraic one, and we
prove the uniqueness of the geometric solutions. Then we do explicit
calculations for three interesting examples. With the first two examples we see
that not all algebraic solutions are geometric and that the algebraic solutions
are not unique. The third example is a non-hyperbolic manifold that admits a
positive, partially flat solution of the compatibility and completeness
equations.
