A possible first step to try to generalise our findings is to take a line that does not go through the origin.
Think about what was special about a line through the origin. Why did this make it a good starting point?
If we choose a line such as \(y = x +1\) then we can draw points such that the \(x\) intercepts differ by \(2\) and the \(y\) intercepts differ by \(3\).
If we drew a line through these two points, would the two lines satisfy the required conditions?
By choosing \(y = x +1\), we cannot modify the gradient or the intercepts of this line. Instead, perhaps it would be more useful to focus on one aspect. We could pick a point that the line passes through, or we could decide on the gradient of the line.
To explore a more general approach and get a feel for what is happening you can use the GeoGebra applet below or use this Desmos graph. You can click on the circles on the left to hide and show different lines as you wish.
Why are there four other lines along with the initial black line? Do these cover all possibilities? Could there be any other lines we should consider?
Below are discussions of three different approaches.