The problem of solving a system of linear ordinary differential equations with constant coefficients is one of the most important problems in both the theory of ordinary differential equations and linear algebra. Therefore, on the one hand, new methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one unknown function.In this paper, we consider a refinement of the method for reducing a system of linear ordinary differential equations with constant coefficients to a single higher-order equation, which makes it possible to find a general solution to the original system; namely, we study the expressibility of all functions of the system of linear homogeneous differential equations with constant coefficients x^' (t)= A⋅x(t) in the form of linear combinations of derivatives of only one unknown function x_k (t), which is part of this system. For any matrix A, all of whose eigenvalues are not multiples, a new simple criterion for expressibility in terms of matrix ranks is formulated, and its correctness is proved. The result obtained can also be applied in the study of solutions of the system x^' (t)= A⋅x(t) for periodicity and in the study of linear systems for complete observability.