Linearly averaged position moment (LAPM) [Rk11Rk22⋅⋅⋅ Rk3N3N] is defined as the linear (not quadratic) average of the position moment operator Rk11Rk22⋅⋅⋅ Rk3N3N over the N-electron wave function Ψ({Rj}), where {Rj} are 3N Cartesian coordinates of electrons and {kj} are non-negative integers. When all the LAPM’s are well defined, it is shown that the Schrödinger equation is equivalent to a set of an infinite number of equations between LAPM’s involving the potential-energy operator. The kinetic energy operator does not appear. The spherical polar representation of the LAPM equation is also presented. Illustrations are given for simple one- and two-electron atoms, where the LAPM equation is applied to the determination of approximate wave functions and associated energies.