This paper is focused on error-free solution of dense linear systems using residual arithmetic in hardware. The designed Modular System uses hardware identical Residual Processors (RP)s for solving independent systems of linear congruences and combines their solutions into the solution of the given linear system. This approach uses the residue number system which is based on the Chinese remainder theorem. In order to efficiently exploit parallel processing and cooperation of the individual components, a hardware architecture of the Modular System with several RPs is designed. In order to verify the proposed architecture, a Xilinx FPGA with a MicroBlaze processor was used. Experimental results are obtained for an evaluation FPGA board with Virtex 6. Results from implementation serve for subsequent theoretical analysis of the system performance for various linear system sizes and further improvement of the system. The proposed system can be useful as a special hardware peripheral or a part of an embedded system for solving large nonsingular systems of linear equations with integer, rational or floating-point coefficients with arbitrary precision.