Challenge Level

Solve the two pairs of simultaneous equations:

\begin{eqnarray} x + 0.99999y & = & 2.99999 \\ 0.99999x + y & = & 2.99998 \end{eqnarray} and \begin{eqnarray} x + 1.00001y & = & 2.99999 \\ 0.99999 x + y & = & 2.99998. \end{eqnarray}

Explain why the solutions are so different and yet the pairs of equations are nearly identical.

NOTES AND BACKGROUND

In this question a small perturbation in one of a pair of equations makes a big change in the solutions. Considering the geometrical properties of the lines represented by the equations helps to de-mystify the results.