The potential of integrating public transit with ridesharing includes shorter travel time for commuters and higher occupancy rate of personal vehicles and public transit ridership. In this paper, we describe a centralized transit system that integrates public transit and ridesharing to reduce travel time for commuters. In the system, a set of ridesharing providers (drivers) and a set of public transit riders are received. The optimization goal of the system is to assign riders to drivers by arranging public transit and ridesharing combined routes subject to shorter commuting time for as many riders as possible. We give an exact algorithm, which is an ILP formulation based on a hypergraph representation of the problem. By using the ILP and the hypergraph, we give approximation algorithms based on LP-rounding and hypergraph matching/weighted set packing, respectively. As a case study, we conduct an extensive computational study based on real-world public transit dataset and ridesharing dataset in Chicago city. To evaluate the effectiveness of the transit system and our algorithms, we generate data instances from the datasets. The experimental results show that more than 60% of riders are assigned to drivers on average, riders' commuting time is reduced by 23% and vehicle occupancy rate is improved to almost 3. Our proposed algorithms are efficient for practical scenarios.