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For a relatively new class of linearization of implicit constitutive relations, wherein the linearized strain tensor is assumed to be a function of the Cauchy stress tensor, we write the balance of linear momentum and the time differentiated constitutive relation as a first order system, and we examine conditions for the hyperbolicity of such a system; this procedure is carried out for one and three dimensions. For the one dimensional case we use the characteristic polynomial and find conditions so that our system is hyperbolic. For three dimensions we find conditions so that our system can be put in a symmetric hyperbolic form. • We write the equations as a first order system. • For the one dimensional case we use the characteristic polynomial and find conditions so that our system is hyperbolic. • For three dimensions we find conditions so that our system can be put in a symmetric hyperbolic form. [ABSTRACT FROM AUTHOR]