Blind source separation (BSS) aims to discover the underlying source signals from a set of linear mixture signals without any prior information of the mixing system, which is a fundamental problem in signal and image processing field. Most of the state-of-the-art algorithms have independently handled the decompositions of mixture signals. In this paper, we propose a new algorithm named multi-task sparse model to solve the BSS problem. Source signals are characterized via sparse techniques. Meanwhile, we regard the decomposition of each mixture signal as a task and employ the idea of multi-task learning to discover connections between tasks for the accuracy improvement of the source signal separation. Theoretical analyses on the optimization convergence and sample complexity of the proposed algorithm are provided. Experimental results based on extensive synthetic and real-world data demonstrate the necessity of exploiting connections between mixture signals and the effectiveness of the proposed algorithm.