There are two different kinds of power-ups you can make use of in The Impossible Line - the miner's hat and the reveal. You will get more use out of the miner's hat, as long as you take things very slowly and listen out for the proximity buzzer. Reveals should only be used in the most complex levels that are found towards the end of the game.
Keep moving the chalk in one continuous line while the memory of the layout is fresh. If you stop, you increase the chances that you'll forget how the maze is laid out.
Listen out for the proximity beeper at all times. This noise indicates that you are getting too close to a wall for comfort, and the more rapid the beeping noise becomes, the closer you're getting to failing the level!
Keep an eye out for any parallel lines at play in the map you're working on. If you know that there are parallel lines right next to each other, then you can use your existing chalk line as a guide for the next line you have to draw.