In this paper, the schemes of approximation of systems with delays by special systems of ordinary differential equations are considered and the connections between their solutions are investigated. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. We consider using approximation schemes to approximate location the asymptotic roots of quasipolynomials of linear systems of differential-difference equations with many delays through the roots of the characteristic equations of the corresponding approximating systems of ordinary differential equations. An algorithm for studying the stability of systems of linear differential-difference equations with many delays is proposed. Numerical algorithms for studying the stability of linear stationary systems with delay are constructed and their coefficient regions of stability are modeled. The implementation of the proposed algorithms for studying the stability of solutions of linear differential equations with delay is demonstrated on model test examples. The analysis of numerical experiments carried confirms the theoretical results presented in the paper.