  In this paper we describe a new method of obtaining ideal solutions of the
well-known Tarry-Escott problem, that is, the problem of finding two distinct
sets of integers $x_i,\;i=1,\,2,\,\dots,\,k+1$ and
$y_i,\;i=1,\,2,\,\dots,\,k+1$ such that $ \sum_{i=1}^{k+1} x_i^r =
\sum_{i=1}^{k+1} y_i^r,\;\;\;r = 1,\,2,\,\dots,\,k$, where $k$ is a given
positive integer. When $k > 3$, only a limited number of parametric/ numerical
ideal solutions of the Tarry-Escott problem are known. In this paper, by
applying the new method mentioned above, we find several new parametric ideal
solutions of the problem when $k \leq 7$. The ideal solutions obtained by this
new approach are more general and very frequently, simpler than the ideal
solutions obtained by the earlier methods. We also obtain new parametric
solutions of certain diophantine systems that are closely related to the
Tarry-Escott problem. These solutions are also more general and simpler than
the solutions of these diophantine systems published earlier.
