I have an HP Pavilion g7-2069wm Notebook PC (64bit) with 6gb ram and a 750 GB hard drive. I was dual booting Ubuntu 14.04 (sda1) and Linux Mint 17.2(sda6). I decided to make Linux Mint my only OS so I used Gparted to delete the sda1 partition. After that I could not boot the SDA6 Linux Mint it would go so far then quit. It drops to a shell with the following alert "/dev/disk/by-uuid/638d0559-6a99-4ddc-9219-d0da6a4d5d92 does not exist. Dropping to a shell":
BusyBox v1.21.1 (Ubuntu 1;1.21.0-1ubuntu1) built-shell (ash)
Enter 'help' for a list of built-in commands.
(initramfs) _

I then installed linux Mint 17.2 on SDA1 and it boots fine. SDA6 (the old mint installation) shows up and I can navigate through all the files and directories with no problem. Just can't get it to boot.
I have looked at all the boot logs and it doesn't seem to log the sequence. I might not be looking at the right logs. I looked at the boot.log, bootstrap.log, syslog.1 but did not find where the boot sequence actually quit.
Any help would be greatly appreciated.

That's the UUID for one of your partitions so open a terminal and run this command: sudo blkid
Take a look at the output to see if the UUID above shows up.

Why delete the partition, why not just format it?

You could get the boot repair script at the link below and select the option to "Create BootInfo summary" and post a link to the output here. It will have details on drives/partitions, boot files and more which should provide more information so someone can assist you.

https://help.ubuntu.com/community/Boot-Repair

The entry is probably in fstab, on sda6. Comment out the line for that disk and it should boot.

The boot loader and /etc/fstab need to have the information to locate the drive. The /dev/ names can be unpredictable since most things have more than one drive these days. So the line in /etc/fstab for / should be UUID=. And then line in /etc/boot/grub/grub.cfg that points at the kernel should be have root=UUID= and the appropriate string which update-grub tends to pull from /etc/fstab. Depending on your version of things. So as previously said, the /etc/fstab probably isn't correct or missing. Also under systemd, if /etc/fstab is missing it will boot it just fine, but / will be read-only by default. If /etc/fstab is present and correct, then / will be read-write (in a favorable wind).

It may be possible to also make the entries generic.