The windows 10 "digital entitlement" is tied to the specific computer when you upgrade it from an earlier version of windows or activate Win10 on that computer with a win 7/8/8.1 key. There is no way to transfer that specific digital license to another computer.
If you have a windows 7/8/8.1 key for your old computer, you can use that to activate windows 10 on your new computer. If you don't have a key you'll need to pay the $22 for your new computer.
Your old system should still be fully activated-- install Win10 and skip inputting the key during the install, boot it up with no key at all, connect to the internet, and it should show as activated.