A neural circuit to solve a system of simultaneous linear equations is presented. The circuit employs non-linear feedback to achieve a transcendental energy function that ensures fast convergence to the exact solution while enjoying reduction in hardware complexity over existing schemes. A new building block for analog signal processing, the digitally controlled differential voltage current conveyor (DC-DVCC) is introduced and is utilized for the non-linear synaptic interconnections between neurons. The proof of the energy function has been given and it is shown that the gradient network converges exactly to the solution of the system of equations. PSPICE simulation results are presented for linear systems of equations of various sizes and are found to be in close agreement with the algebraic solution. The use of CMOS DC-DVCCs and operational amplifiers facilitates monolithic integration.