In the realm of Hilbert space, the applicability of the system of modeling equations in Operator Theory is investigated and considered. Recently, the interest in studying this type of systems has increased greatly. This article, employing the operators’ equations applicable to Hilbert space, deals with two systems of equations. It attempts to examine the required stipulations for the presence of a solution to this posed system. The first system comprises of two equations. For Λ and F that achieve regularity with non-invertibility, the general solution formula is investigated based on the Moore-Penrose inverse. Furthermore, the second system includes four equations having adjoint operators. Among the required stipulations, the operator attained self-adjoint attribute. The general solution formula is established based on the are regular of the operator when both Λ and F are regular.