Let

and consider the line . Show that each point on satisfies the system of linear equations:

*Proof.* First, we solve the system of equations. (I’m using matrices to do this since it’s faster and easier. Apostol may not have actually covered this yet, but you can solve it by eliminating variables in basically the same way.) We have,

There, the variable is free and we have , . Letting we have

But these are exactly the points