I kinda figured this was a newer computer and components. What video card do you have in this machine... how old is it(the card)? do you have another machine that you could put it in to see if the behavior occurs in that machine? If so I would suggest contacting the mfg to see if there is a driver issue or something they would suggest.
How about your monitor? Is it new? Does it display the line if connected to another machine?
This is understandably annoying but resolvable.