Two disks are backed up when doing a system backup. One disk is Win7, the other is Win10. Win 7 is the system disk. However, the GOAL is to backup Win10 only. Disconnecting the Win7 disk, the Win10 does not boot, therefore, does not change to become the system disk to image. Apparently it is missing the MBR. Exactly, how is this CORRECTED so that a system backup of the Win10 disk can be executed?The boot area to select the version of Windows you want to run is on the Windows 7 drive in a small partition NORMALLY 100 mb. So you need to include that partition in any backup.
What software are you USING to do the backup?

Also why are you only backing up the Windows 10 drive?
The software is AOMEI Backupper Professional which hasn't been allowing for a backup of Win7 & Win10 individually i.e. it insists on one (large) image for both OSes. AOMEI Support states nothing can be done about it. Win7 readily backs up other partitions that are bootable without having to include Win10 in the image file. It's always when imaging Win10 that this issue is confronted.