A two-car crash with sticking

Suppose two cars collide, stick together, and skid off together. If we know the cars' initial momentum vectors, we can count equations and unknowns as follows:

unknown #1: x component of cars' final, total momentum

unknown #2: y component of cars' final, total momentum

equation #1: conservation of the total p_x

equation #2: conservation of the total p_y

Since the number of equations equals the number of unknowns, there must be one unique solution for their total momentum vector after the crash. In other words, the speed and direction at which their common center of mass moves off together is unaffected by factors such as whether the cars collide center-to-center or catch each other a little off-center.