A system of differential equations with linearly transformed arguments of the form da d¿ = X(¿; a¤; '£); d' d¿ = !(¿ ) " + Y (¿; a¤; '£); (1) where ¿ 2 [0;L], a 2 D – limited area in Rn, ' 2 Rm, small parameter " 2 (0; "0], "0 ¿ 1, a¤ = (a¸1 ; : : : ; a¸p ), '£ = ('µ1 ; : : : ; 'µq ), 0 < ¸1 < ¢ ¢ ¢ < ¸p · 1, 0 < µ1 < ¢ ¢ ¢ < µq · 1, a¸i (¿ ) = a(¸i¿ ), 'µj (¿ ) = '(µj¿ ) is investigated. Vectorfunctions X and Y are defined and smooth enough for all variables in the area G = [0; ¿ ] £ Dp £ Rqm and are 2¼-periodic by vector components '£. For the system (1) the following conditions are set (1) Xr º=1 ®ºa(xº) = Z¿2 ¿1 f(¿; a¤; '£)d¿; Xr º=1 ¯º'(xº) = Z¿2 ¿1 g(¿; a¤; '£)d¿; (2) where 0 · x1 < x2 < ¢ ¢ ¢ < xr · L, 0 · ¿1 < ¿2 · L. In [1], for the system (1) under conditions (2), a much simpler problem is constructed by averaging [1] the vector functions X, Y , f and g over fast variables 'µº on the cube of periods [0; 2¼]mq. The existence and uniqueness of the solution of the problem (1)-(2) in the class C1[0;L] is proved. On the time interval [0;L], some estimates of the deviations of the order mq p " between the solutions of the problem (1)-(2) and the solutions of the averaged system are established. The results of the work generalize the results obtained in [2].