The plane problem of linear elasticity theory is considered in Cartesian coordinates for a nonequilateral trapezoidal anisotropic region with arbitrary side geometry. The concepts of static and geometric problem unknowns are introduced with infinite-series expansions in Legendre polynomials of general form and arbitrary interval. A dual form of notation is introduced for the arbitrary unknown functions of the problem in accordance with the boundary conditions on the sides of the trapezoid. A process is constructed for reduction of the equilibrium equations and kinematic relations of the plane problem to ordinary first-order differential equations using Legendre polynomials of general form. Identical satisfaction of static and kinematic boundary conditions on the sides of the trapezoid is ensured in the reduction process. The resulting regular infinite sequence of systems of ordinary first-order differential equations and the corresponding boundary conditions for the coefficients in the series expansions of the unknowns is identical to the initial equations of plane elasticity theory.