In this paper, we introduce a modified viscosity subgradient-like extragradient implicit rule with line-search process for finding a solution of a general system of variational inequalities (GSVI) with a variational inequality (VIP) and a fixed-point (FPP) constraints in Hilbert spaces. The suggested algorithms are based on the subgradient extragradient method with line-search process, hybrid Mann implicit iteration method, and composite viscosity approximation method. Under suitable restrictions, we demonstrate the strong convergence of the suggested algorithm to a solution of the GSVI with the VIP and FPP constraints, which is a unique solution of a certain hierarchical variational inequality.