Animate the drawing of a line across the screen

I currently use Flash CS5. I am attempting to demonstrate the flow of control oil through a series of hydraulic actuating valves. I want to have a thin, black line (stroke--1 pt) representing the oil “draw itself” across the screen. The line will extend in a convoluted path horizontally and vertically through a series of valves before terminating in a servo cylinder.

I can make this work using a shape tween, but to extend or draw the line, I have to make two copies of the line (a short line and then a longer line). And I have to do this every time the line changes direction—every time it turns horizontally or vertically.

Is there is a more efficient way of doing this?

Thanks