OS System on old macbook air

1. Start up from macOS Recovery

To start up from macOS Recovery, turn on your Mac and immediately press and hold one of the following combinations on your keyboard. Command-R is generally recommended, especially if you never installed macOS Sierra 10.12.4 or later… https://support.apple.com/en-us/HT204904#notupdated

Command (⌘)-R

Install the latest macOS that was installed on your Mac.

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Shift-Option-⌘-R

Install the macOS that came with your Mac, or the closest version still available.

2. Decide whether to erase (format) your disk

If you need to erase your disk before installing macOS, select Disk Utility from the Utilities window, then click Continue. You probably don't need to erase, unless you're selling or giving away your Mac or have an issue that requires you to erase. Learn more about when and how to erase.

3. Install macOS

After starting up from macOS Recovery, follow these steps to install macOS:

1. Choose Reinstall macOS (or Reinstall OS X) from the Utilities window.
2. Click Continue, then follow the onscreen instructions to choose your disk and begin installation.
3. If the installer asks to unlock your disk, enter the password you use to log in to your Mac. If it 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. 

1. Please allow installation to complete without putting your Mac to sleep or closing its lid. During installation, your Mac might restart and show a progress bar several times, and the screen might be empty for minutes at a time.

If your Mac restarts to a setup assistant, but you're selling or giving it away, press Command-Q to quit the assistant without completing setup. Then click Shut Down. When the new owner starts up the Mac, they can use their own information to complete setup.

If you never installed macOS Sierra 10.12.4

If macOS Sierra 10.12.4 or later was never installed on your Mac, macOS Recovery works differently:

• Command-R is still the recommended way to start up from macOS Recovery. This combination makes sure that the installation isn't associated with your Apple ID, which is important if you're selling or giving away your Mac.
• Option-Command-R installs the macOS that came with your Mac, or the closest version still available.
• Shift-Option-Command-R isn't available.

https://support.apple.com/en-us/HT204904