Polyelectrolyte solutions are modeled as freely tangent-jointed, charged hard-sphere chains and corresponding counterions in a continuum medium with permitivity E. By adopting the sticky-point model, the Helmholtz function for polyelectrolyte solutions is derived through the r-particle cavity-correlation function (CCF) for chains of sticky, charged hard spheres. The r-CCF is approximated by a product of effective nearest-neighbor two-particle CCFs: these are determined from the hypernetted-chain and mean-spherical closures (HNC/MSA) inside and outside the hard core, respectively, for the integral equation theory for electrolytes. The colligative properties are given as explicit functions of a scaling parameter Gamma that can be estimated by a simple iteration procedure. Osmotic pressures, osmotic coefficients, and activity coefficients are calculated for model solutions with various chain lengths. They are in good agreement with molecular simulation and experimental results.