## Task

Lisa is working with the system of equations $x + 2y = 7$ and $2x - 5y = 5$. She multiplies the first equation by 2 and then subtracts the second equation to find $9y = 9$, telling her that $y = 1$. Lisa then finds that $x = 5$. Thinking about this procedure, Lisa wonders

There are lots of ways I could go about solving this problem. I could add 5 times the first equation and twice the second or I could multiply the first equation by -2 and add the second. I seem to find that there is only one solution to the two equations but I wonder if I will get the same solution if I use a different method?

- What is the answer to Lisa's question? Explain.
- Does the answer to (a) change if we have a system of two equations in two unknowns with no solutions? What if there are infinitely many solutions?

## IM Commentary

The goal of this task is to help students see the validity of the elimination method for solving systems of two equations in two unknowns. That is, the new system of equations produced by the method has the same solution(s) as the initial system. This is a subtle and vital point, though students should already be familiar with implementing this procedure before working on this task.

It is not difficult to verify that a solution to the initial system of equations is also a solution to the new system. The key to the success of the elimination method, however, is that all steps in the algorithm are reversible. This is why a solution to the simpler system of equations is also a solution to the original system. This can be seen geometrically when taking a multiple of an equation since, for example, $x + 2y = 7$ and $2x + 4y = 14$ define the same line in the plane. The geometric intuition is lost, however, when equations are added or subtracted as this creates a new line, having the same point of intersection with the line defined by $2x - 5y = 5$.