The aim of the paper is to determine approximations of the interval hull of linear systems of \(n\)-dimensional interval equations, \(Ax=b\). For this purpose, the system is transformed into a system of the form \(x=Cx+b\). The solution set is invariant with respect to this transformation. The latter system is embedded into the space \(\mathbb{R}^{2n}\) and solved by applying a damped Newton algorithm, which is adapted for dealing with nonsmooth functions.