We reformulate and extend the HuPf algorithm (see [7]), which was originally designed for a general 6R manipulator (i.e. 6 jointed open serial chain/robot with only rota-tional joints), to solve the inverse kinematic (IK) problem of 6R/P manipulators (6-jointed open serial robot with joints that are either rotational or prismatic/translational). For the algorithm we identify the kinematic images of 3R/P chains with a quasi-projective variety in P 7 via dual quaternions. More specifically, these kinematic images are projections of the intersection of a Segre variety with a linear 3-space to an open subset of P 7 (identified with the special Euclidean group SE(3)). We show an easy and efficient algorithm to obtain the linear varieties associated to 3R/P subchains of a 6R/P manipulator. We provide examples showing the linear spaces for different 3R/P chains (a full list of them is available in an up-coming paper). Accompanying the extended HuPf algorithm we provide numerical examples showing real IK solutions to some 6R/P manipulators.