The relaxation approximation for systems of conservation laws has been
studied intensively for example by [17, 5, 19, 24]. In this paper the corresponding
relaxation approximation for 2x2 systems of balance laws is studied. Our driving
example is gas flow in pipelines described by the isothermal Euler equations. We
are interested in the limiting behavior as the relaxation parameter tends to zero. We
give conditions where the relaxation converges to the states of the original system
and counterexamples for cases where the steady states depend on the space variable.