We concern the computation of approximate mixed Nash equilibria in symmetric weighted congestion games, which has been shown to be PPAD-complete. We focus our discussion only on affine linear latency functions, and propose an algorithm deriving from the best response dynamics. Our algorithm efficiently computes an \(\epsilon \) -approximate mixed Nash equilibrium within a polynomial runtime parameterized mainly by the maximum player weight W, where \(\epsilon \in (0,1)\) is an arbitrary small constant. This then provides the first polynomial runtime algorithm for computing an \(\epsilon \) -approximate mixed Nash equilibrium in a weighted congestion game, though the players are programmed to have the same strategy set, the latency functions are assumed to be affine linear, and the polynomial runtime is still parametric.