piecewise and non-piecewise isomorphic?

Ok, let me give the problem first:

Quote:

Consider the following interpretation of these terms: "points" are ordered pairs (x,y) in the xy-plane, "lines" are the functions

$\displaystyle

y = \begin{cases}

mx+b & x < 0 \\

2mx+b & x \ge 0

\end{cases}

$

And say a "point" "lies on" a "line" if the point (x,y) satisfies the equation. Show that this interpretation is a model for incidence geometry and determine which, if any, parallel property it has.

When visualizing the problem, I usually imagine the y-axis is the surface of a body of water, and the lines are light rays being refracted.

I'm already nearing a solution for the problem. However, there is one related thing I'd like to do for my own edification: prove that $\displaystyle

y = \begin{cases}

mx+b & x < 0 \\

2mx+b & x \ge 0

\end{cases}

$ is isomorphic with $\displaystyle

y = mx+b

$ Intuitively it seems true, but I don't have a clue as to how to go about proving it. Can anyone provide me with such a clue? (Thinking)

Notes:

*If anyone wants to see my solution to the actual problem (once I finish it) let me know and I'll make an edit to this post.**The actual problem says "undefined terms" instead of "these terms" in the first sentence.*