This work presents a novel algorithmic framework, called Fast Adaptive Alternating Direction Method of Multipliers with Gaussian Back Substitution (ADMM-G-V), tailored for solving multiple block linear constrained separable problems. The proposed method extends the classical multi-block ADMM by incorporating an adaptive penalty parameter, which is dynamically adjusted during the iterative process to enhance convergence properties and computational efficiency. A comprehensive theoretical analysis is provided by establishing the global convergence and worst-case convergence rate of the algorithm in both ergodic and non-ergodic senses. We demonstrate the effectiveness of our method through numerical experiments on consensus problems over networked agents and distributed logistic regression tasks.