When we write

α

for 0.99999 in the given equations we get:

\begin{matrix} x + α y & = 2 + α & (1) \\ α x + y & = 1 + 2 α & (2) \end{matrix}

and

\begin{matrix} x + (2 - α) y & = 2 + α & (3) \\ α x + y & = 1 + 2 α . & (4) \end{matrix}

Solving these equations the first pair has the solution

(x, y) = (2, 1)

and the second pair has solution

(x, y) = (\frac{- 2 α}{1 - α}, \frac{1 + α}{1 - α}) .

1 - α

is small (

10^{- 5}

), the solutions of the second pair of equations are large.

(x, y) = (- 199998, 199999)

to the nearest whole number.

The reason that the solutions to the pairs of simultaneous equations are so different is that the two lines represented by the pairs of equations are nearly parallel in both cases (both being close to the line $x + y = 3$ ). As the lines are so close together, when one line is rotated slightly there is a big change in the point of intersection from (2,1) to (-199998, 199999)