S. Mardešić posed the question: Given a Čech system \({\mathbf C}(X) = (|N({\mathcal U})|, [p_{{\mathcal U}{\mathcal V}}], Cov(X))\) of a topological space \(X\), is it possible to find representatives \(q_{{\mathcal U}{\mathcal V}}\) of the homotopy classes \([p_{{\mathcal U}{\mathcal V}}]\) so that the system \((|N({\mathcal U})|, q_{{\mathcal U}{\mathcal V}}, Cov(X))\) is an approximate system? Here the approximate system is in the sense of S. Mardešić and L. R. Rubin [Pac. J. Math. 138, No. 1, 129–144 (1989; Zbl 0631.54006)]. In this paper, the author answers the question in the negative by showing that for any Hausdorff arc-like continuum \(X\), the Čech system \({\mathbf C}(X)\) of \(X\) does not induce any associated approximate system.