I am not sure why that is. The bootable version is based on linux. I would then use a Gparted Live CD. All steps are virtual just as Partition Wizard is but you need to select the drive you are working on in the dropdown box in the upper right of the window.
Here is a good tutorial on it's use. The operation will be the same as in my previous post and you can move the partition bar by left clicking and dragging.All steps are virtual until you select Apply. If Gparted does not work then I would post this question in the Linux Forum and reference this thread. There also a number of youtube videos on it's use.
When you say the linux partition cannot be moved, this is the partition next to unallocated space, correct? Another thing, is the linux partition listed as logical? It may be part of an extended partition than included the Linux System Partition. Gparted should see the Extended Partition. Click the Extended partition that includes both the Linux System Partition and the logical Linux Partition.
In the below image you can see that partition sda2 is extended that includes the logical partitons of sda5, sda6, and sda7. When you start Gparted see if you have an extended partition. Click the extended partition and see if you can move it which would include your Linux System Partition and Linux Partition. I have never done this so I am not sure it will work.
Edit: I see you do indeed have an extended partition. Boot Gparted Live CD and select your disk in the dropdown box. Click on your extended partition and see if you can move it which includes your Linux System, and Swap.
Edit Edit: If you do decide to reinstall linux you do not need to delete your Windows partition. When you reinstall linux it should see your Windows partition and create the Grub Bootloader again. If you can't boot you can use a Boot Repair CD that will automatically repair Grub.
Edited by JohnC_21, 11 November 2015 - 09:54 PM.