A gear shifting system is described that includes an first shaft that is connected to a motor to be rotated thereby, a synchronizing gear, a first insert unit and a second insert unit protrude at both sides thereof, a first gear and a second gear configured to rotate around the first shaft, a first plate and a second plate, a first engagement aperture and a second engagement aperture, and a second shaft. The second shaft is configured to move the synchronizing gear in an axial direction so that the first insert unit is inserted into the first engagement aperture or the second insert unit is inserted into the second engagement aperture to transfer a torque of the synchronizing gear to the first gear or the second gear.