Cant reinstall the os system on my older Mac air
Having hard time resetting my Mac air
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Perhaps instructions and ideas given in these articles may be of some use.
• About macOS Recovery on Intel-based Mac computers - Apple Support
https://support.apple.com/en-us/HT201314
[This also includes an alternate instruction ^ should 'internet recovery' not work.]
• How to reinstall macOS - Apple Support
https://support.apple.com/en-us/HT204904
IF the installer doesn't see your disk, or it says that it can't install on
your computer or volume, you might need to erase your disk first.
There are other options about using different installers with this instruction set.
• How to get old versions of macOS - Apple Support
https://support.apple.com/en-us/HT211683