You're right, Johnathan, unless, of course, the old computer was sold sans the Win 7 that came installed on it. In that case, it might be possible to use the Win 7 installation if a call to MSFT deems it appropriate. I doubt such would occur but one never knows unless they try.
If the old computer was sold with Win 7 as the OS then, obviously, the OP needs to buy a new copy and the aforementioned scenario becomes irrelevant.