This paper proposes a method combining external modulation technology and master-slave injection-locking technology to obtain a broadband linear-chirped laser source, and the laser source is used to realize a frequency modulated continuous wave light detection and ranging (LIDAR) system for simultaneous distance and speed measurement. The master-slave injection-locking system includes a narrow line-width fiber laser as the master laser and a distributed feed-back (DFB) semiconductor laser as the slave laser, and their wavelengths are both about 1550 nm. A triangular linear-chirped microwave signal with a frequency range of 8--14 GHz is used to modulate the output from the master laser, and then the modulated laser is injected into the DFB semiconductor laser to obtain a broadband linear-chirped laser with a 6 GHz bandwidth and a side-mode suppression ratio over 20 dB. We build a frequency modulated continuous wave LIDAR based on the laser source. Using this system, we perform the distance measurement of static objects and the instantaneous measurement of distance and speed of a moving object. The distance and speed of a moving object are obtained by different formulas and verified with each other. The distance and speed resolutions are 2.5 cm and 7.8 mm/s, respectively.