Consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation. This requires to decompose the second-order derivative terms of the velocity into first-order ones. If the Maxwell laws are concerned, the decompositions lead to approximate systems with scalar, vector and tensor variables. We construct approximate systems without tensor variables by using Hurwitz-Radon matrices, so that the systems can be written in the standard form of symmetrizable hyperbolic systems. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global convergence in time holds if the initial data are near constant equilibrium states. We also prove the convergence of the approximate systems with tensor variables.