A.M. Samoilenko’s numerical-analytic method is a well-known and effective research method of solvability and approximate construction of the solutions of various boundary value problems for systems of differential equations. The investigation of boundary value problems for new classes of systems of functional- differential equations by this method is still an actual problem. A boundary value problem for a system of differential equations with finite quantity of transformed arguments in the case of linear two-point boundary conditions is considered at this paper. In order to study the questions of the existence and approximate construction of a solution of this problem, we used a modification of A.M. Samoilenko’s numerical-analytic method without determining equation, i.e. the method has an analytical component only. Sufficient conditions for the existence of a unique solution of the considered boundary value problem and an error estimation of the constructed successive approximations are obtained. The use of the developed modification of the method is illustrated by concrete examples.

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