Problem $2(20\%)$

Assume an origin is connected to a destination with two routes, where both routes have users.

Assume the travel time of each route has a linear relationship with the traffic flow on the route,

i$.$e$.$

$t_1=a_1+b{x}_1$;$t_2=a_2+b{x}_2$;

Prove that if $a_1=a_2,$ then the User Equilibrium and System Optimum Solutions are the same.