I am admin, but I cannot edit system variables (Windows 10)

If accessing this through the System Page

Windows + Pause/Break will bring up the System Page

Then click Advanced system settings

This might get you to auth at this point - but even if it doesn't, it should ensure you're continuing as administrator

Click the Environment Variables button at the bottom

You should now be on the environment variables page as an Admin.

If accessing through start menu:

(Just noticed the other answer by Niall895 is pretty much the same as below)

When you type 'environment' into the start menu you will see two similar options:

Edit the system environment variables.

Edit environment variables for your account.

These may appear in a different order to the screenshots below, depending on which one you opened last. Make sure you select the shortcut with system in it:

Edit the system environment variables.

If you don't choose this option, the system variables buttons New..., Edit... and Delete will be greyed out as in your screenshot.

(It may or may not get you to re-type your credentials first)

It's possible you have different UAC settings on this instance

(Type UAC into start menu on each and look at the settings)

Although changing this just usually gets rid of the prompt pop-ups