Intersections
Problem
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.
Getting Started
In this case, rather than working with the awkward numbers, you may find it easier to substitute $\alpha = 0.99999$ and solve the equations algebraically, and finally substitute $0.99999$ for $\alpha$ to find the numerical solutions.
To explain the solutions think about the geometry associated with these equations.
Student Solutions
We had good solutions from Andrei in Bucharest, and from Matt and Andrew at the Perse School
The solution in the first case is $x = 2, y = 1$ and in the second case is $x = - 199998, y = 199999.$
For the first two lines one gradient is a little under minus one and the other gradient a little over.
For the second two lines both gradients are a little above minus one.
All four lines cut the $y$-axis very near to $3$.
Because the pairs of lines in each case are nearly parallel the slight and unique change in each line's gradient away from minus one causes the intersection to occur in very different places.
Teachers' Resources
Why do this problem?
On the one hand this problem is a simple exercise in solving pairs of linear simultaneous equations but on the other it provides perhaps unexpected results that call for investigating the connection between the algebra and geometry, and considering the equations of the lines and gradients.
Possible approach
Set this as homework or as a lesson starter and have a class discussion about the results.
Key questions
Why are the solutions of the two pairs of simultaneous equations so different when the equations are so nearly the same?