This paper deals with the two dimensional Cauchy problem: \[ \partial u_ j/\partial t+\lambda_ j(\partial u_ j/\partial x)=\epsilon f_ j(u_ 1,...,u_ n) \] \[ u_ j(0,x,\epsilon)=u_{j0}(x)+(\sum^{k}_{i=1}\epsilon^ iu_{ji}(x))+\epsilon^{k+1}u_{jk+1}(x,\epsilon) \] involving a small parameter \(\epsilon >0\) and real numbers \(\lambda_ 1,...,\lambda_ n\). Assuming a series development in powers of \(\epsilon\) for the solutions in the form \[ u_ j(t,x,\epsilon)=v_{j0}(\tau,y_ j)+\sum^{\infty}_{i=1}\epsilon^ i(v_{ji}(\tau,y_ j)+w_{ji}(\tau,t,x)) \] where \(\tau =\epsilon t\) and \(y_ j=x- \lambda_ jt,\) Cauchy problems to determine \(v_{ji}\) and \(w_{ji}\) are formulated. The unique local solvability of these new Cauchy problems in appropriate function spaces is established under suitable hypotheses on the functions \(f_ j\) and members \(\lambda_ j\) (1\(\leq j\leq n)\) and on the Cauchy data of the original problem. The error introduced by truncating the series for \(u_ j\) with its \((k+1)th\) term is estimated.