When do you use etre a instead of la mien, la tien etc?

Bonjour Bernadette,
Very interesting question! The expression 'être à' means literally 'to be to <someone>', where 'le mien / le tien...' mean 'mine/yours...'.
Therefore they are synonymous, although you don't use them in the same way:
e.g. Ce livre est à moi. (This book is mine.)
C'est le mien. (It's mine)

You can also have a look at our two related lessons:
https://progress.lawlessfrench.com/revision/grammar/how-to-express-mine-yours-his-hers-le-mien-le-tien-le-sien-etc-singular-possessive-pronouns
https://progress.lawlessfrench.com/revision/grammar/how-to-express-possession-with-etre-a

A bientôt !