A unified affine-projection-like adaptive (UAPLA) algorithm is deivised and verified for system identification. The UAPLA algorithm uses a generalized cost function encompassing some data-reusing methods to cope with colored input signals. Furthermore, the UAPLA algorithm is derived based on the new cost function to generalize multiple affine-projection algorithms. As a result, the proposed UAPLA algorithm includes some classical adaptive filters as exceptional cases while allowing flexibility to achieve low estimation errors under impulsive noises. The obtained results conducted by simulations help to corroborate the superiority of the proposed UAPLA algorithm over other popular AP algorithms.