a defective line across the entire LCD display is almost always a sign of a problem with one of the LCD "driver" ICs- since it's a whole line it's unlikely to be the liquid crystal portion of the display itself dying or a problem with the main logic board or graphics controller or a loose connection between the display and logic board.
These LCD driver ICs drive the row and column voltages necessary for the LCD to operate. So a failure in one of the row or column outputs of a driver would show up as a dead line on the display. Sometimes the failure can simply be a mechanical break in one of the many output lines from the driver to the LCD.
the problem is if it is indeed an LCD driver, it may be difficult or impossible for you to repair without replacing the whole LCD display module - this is because the LCD driver ICs are usually very wide (e.g. 120 outputs) bare silicon chips integrated directly on the edges of the glass substrate or bonded to a thin flex substrate along the edge of the panel - practically impossible to manually replace/resolde
to understand better, see this example diagram of an LCD module, consisting of the LCD display itself plus source and gate drivers (row and column drivers) - refer to figure 2 on bottom of this page:
bottom line: if you have a defective line on the display, the problem is almost certainly in the LCD module itself (specifically one of the driver ICs or it's physical connection). the display module will likely need to be replaced.
This could be a number of things. The logic board could have an issue causing anomaly in the screen. The LCD wouldn't be the problem here because when it starts to age pixels die not lines in the screen.
You could take the device apart and clean it. Try making sure all the connections are not loose.
Here is the guide to disassembling it:
If your not up to that take it to apple and see if they can help.