In this paper, we mainly consider operator equations $AX=C$ and $XB=D$ in the framework of Hilbert space. A new representation of the reduced solution of $AX=C$ is given by a convergent operator sequence. The common solutions and common real positive solutions of the system of two operator equations $AX=C$ and $XB=D$ are studied. The detailed representations of these solutions are provided which extend the classical closed range case with a short proof.