- Backup your Windows installation.
- Uninstall Windows from your hard drive.
- Install Windows in a VM.
- Ensure the Windows installation has internet access.
- Attempt to activate the installation. You may need to call Microsoft to complete this step over the phone.
If they ask you why your are re-activating, tell them you have removed the old installation and changed your system configuration and that you are now reinstalling. (All true.)
Don't volunteer that the Windows 7 key is OEM. Your Windows 10 entitlement may or may not be marked as originating from OEM. If they say that it is OEM and that you cannot use it to activate, then you know. Say thanks and end the call. Restore your backup from step 1 and stick with dual boot, or purchase a copy of Windows 10 and activate yours that way.
Of course, you could always install Hyper-V component of Windows 10 and run your Linux inside. No licensing issues that way. And it may be installed and running by default.
Also, before step 2, you might try to find some sort of P2V solution. This basically takes an 'image' of your physical system and converts it to a virtual machine. VirtualBox wiki article. This may trigger a reactivation due to the significant change in the HAL (hardware abstraction layer).