Your file hierarchy lives into two separate partitions: everything in /home
is in a partition with an XFS filesystem. The rest of it (i.e., the root
partition) is in a partition with an ext4 filesystem. This second partition
is almost full because you installed heavy games (a lot of images, musics,
etc. that go into the root partition).
XFS is a great filesystem, which is particularly efficient when it comes to
manage large files (such as those in /home), but it has one very boring
drawback: you cannot reduce it. What you can do, from a Live system, is:
backup (using, for instance, an external disk or the local network and
another computer) its content, i.e., the directories bearing the names of the
users of the system;
delete the XFS partition and the swap partition;
increase the size of the root partition as much as you want (notice that you
do not need a lot much than the current size: because /home is separated, /
only hosts the system that does not grow forever);
recreate, in the extended partition, the swap partition (I do not think you
need it to be as large as currently: when the computer exhausted all the RAM
and starts using the swap, it become so slow that you notice it and start
terminating programs) and the the partition for /home (all the remaining
space; you may want to choose another filesystem if you feel pissed off by
XFS shrinking limitation: ext4 or btrfs if you feel a little bit adventurous:
the very modern btrfs is still marked as "experimental" by kernel
developers);
take note of the number of those two partitions (something like /dev/sda5);
mount the new (and empty) /home filesystem (you can just double click on the
right disk icon in the file browser) and put the back-upped user directories
on it;
mount the extended / filesystem and, in its sub-directory "etc", edit the
file named "fstab". For the swap and /home, you want, in the first column
(instead of the "UUID=...") the /dev/sdX you took note of (X being a
different number for the swap and for /home);
reboot.
Come back here (from the Live system) if you face some troubles. We can help
you. Notice that the important point is to not lose you users' data!
Notice that, if you just want a few GB more for /, then you can just take
them from the swap partition and never delete (only resize) any partition.
