Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show that the decomposed subproblems in the ADM procedure can be substantially alleviated by linearizing the involved quadratic terms arising from the augmented Lagrangian penalty on the model's linear constraints. When the resolvent operators of the separable functions in the objective have closed-form representations, embedding the linearization into the ADM subproblems becomes necessary to yield easy subproblems with closed-form solutions. We thus show theoretically that the blend of ADM, Gaussian back substitution and linearization works effectively for the separable convex minimization model under consideration.