I intend to upgrade from a Windows 7 computer to a new Windows 10 computer.  The Windows 7 system has seven partitions.  I wish to copy the data partitions to the new system.  If I create a Macrium V6 full image backup on the Windows 7 system, create comparable partitions on Windows 10, install Macrium Reflect on the Windows 10 system, can I restore the data files to the Windows 10 system?  I will install the programs manually.

No problem. You have a choice, if your data is on a separate partition on your Windows 7 computer (with no system files) and you have an unused partition or unallocated space on your new Windows 10 computer you can just restore the single data partition, from an Image backup, to the Windows 10 computer. Alternatively you can mount an Image backup in Windows Explorer on your Windows 10 computer and copy and paste your data files OR use file and folder backups.

(a) To restore an Image backup of your Windows 7 computer see the following tutorial: http://knowledgebase.macrium.com/display/KNOW/Restoring+an+image+from+Windows  take care, at step 3 remove the ticks against all the Source partitions and drag only the data partition from the source to the destination in the Windows 10 computer's new data partition (or unallocated space). Click 'Restored Partition Properties' to resize the partition if needed. Do not drag any other partition from Source to Destination, do not click on 'Copy selected partitions'. Make sure you have selected the correct Destination partition to drag the single data partition to from source to Destination. > 'Next'....

(b) For a safer method Mount your Windows 7 Image backup in Windows Explorer and copy and paste the files from the Source to folders in the Windows 10 computer. To Mount the backup in Windows Explorer see: http://knowledgebase.macrium.com/display/KNOW/Browsing+Macrium+Reflect+images+and+backups+in+Windows+Explorer  At step 3 you will need to tick 'Enable Access to restricted folders' near the bottom of dialog.

(c) A third option is to create a File and Folder backup of your data partition of your Windows 7 computer and restore that to Windows 10.
http://knowledgebase.macrium.com/display/KNOW/Backing+up+files+and+folders
http://knowledgebase.macrium.com/display/KNOW/Restoring+a+file+and+folder+backup

If you choose step (a) I strongly suggest you create an Image backup of the whole hard drive of the Windows 10 computer before you start with step (a). If you have restored Images before then (a) should be no problem, it is the method I would choose. I have heard of people choosing the wrong partitions in the source and destination and having problems which is why I suggest you create an Image of your Windows 10 computer so you could recover the whole hard drive if needed.

Please Note: When restoring a single data partition from Windows 7 Image to a Windows 10 data partition it will overwrite the data in the Windows 10 data partition, so if there is already data in this partition then use either (b) or (c) to transfer data into the Windows 10 data partition. Also take care when restoring an Image, as restoring a whole Image backup of the Windows 7 computer to the Windows 10 computer will overwrite the Windows 10 computer and make it a Windows 7 computer so make sure you only put one 'data' partition in the destination.

Hope this helps
Regards, et