For educational purpose, you had better try fixing your installation instead of doing a reinstall. Anyway, during the installation process you can choose a custom partitioning that allows you to specify where the root (/) partition should be. You can select the one used by the previous install (/dev/sda5 if I understand well what you wrote). It would, of course, be formatted. /dev/sda7 would not be touched. As a consequence, you do not need to backup this partition.

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