  In this paper the new procedure for a construction of an approximated
solution to initial data problem for one-dimensional pressureless gas dynamics
system is introduced. The procedure is based on solving the Riemann problems
and tracking singular wave interactions. For that system the new problem with
initial data containing Dirac delta function is solved whenever two waves
interact. Use of the shadow waves as singular solutions to such problems
enables us to easily solve the interaction problems. That permits us to make a
simple extension of the well known Wave Front Tracking algorithm. A
non-standard part of the new algorithm is dealing with delta functions as a
part of a solution. In the final part of the paper we show that the
approximated solution has a subsequence converging to a signed Radon measure.
