There is absolutely no way that you should even think about doing any of this without a complete backup on a separate disk. Verify the backup before beginning.
1) If you have a complete backup, the easiest way to do this would be to wipe everything, make the partitions you want, and then copy the information back from your backup. If your Recovery HD is deleted, there is no way to re-create it without reinstalling the OS. (At least none that I could find when I was searching recently.)
2) Assuming you have a backup OTHER than that Backup partition, if you want to attempt to merge these, I would try it in this order:
a. Delete the 'Backup' partition.
b. Merge 'Data' partition and former 'Backup' partition.
c. Merge System partition and other partition.
d. Re-create the 'Backup' partition and restore it from whatever backup you have.
I have done several live-system partition resizings with no problems. However, the one time that there was a problem, I had to reformat the entire drive and start over.
If I understand your question correctly you are trying to resize the volume group or actually one of the volumes within a group. From what I could gather your disk was converted to a CoreStorage Volume. Could you please verify that by issuing the following command in a terminal and check if you get a similar output to the one in the picture:
diskutil corestorage list
The currently available Disk Utility doesn't support resizing of logical Volumes. However using the command line, there is a undocumented function that allows you to resize Volumes.
If you consult the help command of diskutil you will receive the following output:
chris$ diskutil corestorage
Usage: diskutil [quiet] coreStorage|CS <verb> <options>,
where <verb> is as follows:
list (Show status of CoreStorage volumes)
info[rmation] (Get CoreStorage information by UUID or disk)
convert (Convert a volume into a CoreStorage volume)
revert (Revert a CoreStorage volume to its native type)
create (Create a new CoreStorage logical volume group)
delete (Delete a CoreStorage logical volume group)
createVolume (Create a new CoreStorage logical volume)
deleteVolume (Delete a volume from a logical volume group)
encryptVolume (Encrypt a CoreStorage logical volume)
decryptVolume (Decrypt a CoreStorage logical volume)
unlockVolume (Attach/mount a locked CoreStorage logical volume)
changeVolumePassphrase (Change a CoreStorage logical volume's passphrase)
As you can see, it doesn't offer any option of resizing a Volume, hence my guess why Disk Utility doesn't allow you to resize.
However the following undocumented functions exists:
chris$ diskutil corestorage resizeVolume
Usage: diskutil coreStorage resizeVolume
lvUUID|MountPoint|DiskIdentifier|DeviceNode size
Resize a logical volume, which is one of one or more disks that consume storage
out of a logical volume group. The logical volume group will have more or less
available space after this operation, if it was a shrink or grow, respectively.
Example: diskutil coreStorage resizeVolume
11111111-2222-3333-4444-555555555555 10g
The resizeVolume function isn't listed in the first listing of available commands above but it still exists and it even provides you with an example on how to use it.
With this command and the correct UUID of the Volume it should be possible to resize one of the Volumes within your logical group.
I did some more digging around the command line and came across the following commands:
Physical Volume (Disk) Commands
- resizeDisk (undocumented) – Resize a physical volume
- removeDisk (undocumented) – Remove a physical volume from a logical volume group
- addDisk (undocumented) - Add a new physical volume to a logical
volume group
Logical Volume Commands
- deleteVolume (undocumented) – Delete a logical volume and all of its contents
- resizeVolume (undocumented) – Grow or shrink a logical volume
- resizeStack (undocumented) – Grow or shrink a logical volume as well as the volume group and physical volume.
Best Answer
This is the kind of task I would always have given to iPartition, however, it doesn't like CoreStorage drives [File Vault & Fusion Drives both use this format] & also it will break Drobos.
You also need the current beta before it will support El Capitan.
With all those caveats it makes it harder to recommend these days.