Based on Kruzhkov's theory, N. N. Kuznecov\ [Ž. Vyčisl. Mat. i Mat. Fiz. {\bf 16} (1976), no.~6, 1489--1502, 1627; MR0483509 (58 \#3510)] derived $L^1$ error estimates for suitably regular approximate solutions to the Cauchy problem for the scalar conservation equation $u_t + f(u)_x = 0$. Later his results were extended in several directions by many authors. In particular, in a recent paper B. Cockburn\ and P.-A. Gremaud\ [Math. Comp. {\bf 65} (1996), no.~214, 533--573; MR1333308 (96g:65089)] were able to establish such a priori error estimates without any regularity assumption on the approximate solution. \par Extending the results of Cockburn and Gremaud, the author derives in the present paper similar estimates (without any regularity assumptions on the approximate solution) for a certain model system in dynamic combustion, plus generalizations thereof, originally due to A. Majda. \par $L^1$ a priori error estimates are established for the approximate solution obtained by the viscosity method, as well as for the one obtained by the Engquist-Osher finite difference algorithm. Both these estimates assume that the initial data are elements of $L^1\cap\rm BV$. \par For the viscosity method, a similar estimate for initial data required to lie only in BV is derived in the last chapter of the paper. For this result, however, some regularity of the approximate solution must be assumed.