The paper describes an efficient method for computing the solution x(t) of the system of equations \((A+tB)x=b\). It is shown that \textit{J. H. Wilkinson's} factorization [Recent applications of generalized inverses, Res. Notes Math. 66, 82-99 (1982; Zbl 0491.65025)] provides the Laurent expansion of x(t) for the case when both A and B are singular. The series expansion is justified for the following reason: Given a linear programming problem with the constraints \((A+tB)x=b\), one needs to evaluate only a finite number of terms in the series expansions in order to compare the objective function values for two different basic feasible solutions. The author finally claims that the results of the paper show that the arithmetic complexity of asymptotic linear programming [\textit{R. G. Jeroslow}, Oper. Res. 21, 1128-1141 (1973; Zbl 0283.90030)] is much smaller than was previously known.