If the line is there from the moment you turn the system on to the moment you shut it off, then it's an issue with the display that is not fixable. You will have to replace it.
You can check a site like http://www.parts-people.com to see what a replacement display would cost, unless yours is still under warranty. It's not difficult to replace a display so much as it's involved and a little annoying. Might be worth the extra money if you can find a whole display assembly, including the bezel if you're not comfortable doing these kinds of repairs yourself. And if you're not comfortable taking your system apart at all, I'd just start looking for a new laptop. The display will likely run you in the neighborhood of $150, then there's an hour of labor to have someone put it in, so add probably another $100 to that, which puts you at around the 50% mark of a new system with considerably better specs.