Nowadays, Linear Parameter Varying (LPV) Model Predictive Control (MPC) represents a consolidated approach to optimally regulate multivariable nonlinear systems imposing constraints on inputs and outputs. The crucial drawback, in particular in embedded LPV-MPC, is represented by the required computational effort. The Quadratic Programming (QP) problem solved by MPC changes at each iteration, and its reconstruction increases drastically the time required to compute the control law. This paper proposes an algorithm to reduce the LPV-MPC computational complexity for a particular class of linear systems. By exploiting a coordinate transformation, the QP reconstruction can be avoided. The effectiveness of the novel method is shown on a benchmark set of random MPC problems, as well as on a real-world case study regarding the control of an heat exchanger cell.