The demodulation part of a traditional fiber grating sensing system with a broadband light source is complex. In order to overcome this shortcoming, a linear swept laser is used as light source of the FBG sensor system. One of the gratings is used as reference grating, and other gratings are used as sensing gratings. The change of the reflection center wavelength of FBG is obtained according to the change of time interval between reflection signals of the sensing grating and the reference grating. The demodulation process is very simple and straightforward. The FBG sensing system is used to accomplish experiments of temperature measurement and fiber axial tension measurement. The results verify the feasibility and the correctness of the FBG sensing system and also prove that the FBG sensing system can either individually or simultaneously measure temperature and fiber axial tension. Some methods which improve the measurement precision are presented, including keeping the temperature of reference grating stable and using the swept light source with low swept rate and the oscilloscope with high sampling rate.