Could Bitcoin be based on accounts instead, more like banks?

Could Bitcoin be based on accounts instead, more like banks?

In banking, a user's account will have a balance, and when they spend an amount is credited from their balance. For example, the account #310 could have $100, and spending $10 would reduce it to $90.

Bitcoin on the other hand makes you spend all $100 from the account in a transaction, but the transaction could have two outputs ($10 and $90 - you would keep control of the new $90 account).

Why was Bitcoin set up that way instead of the other way? The other way seems more natural. You'd just sign a transaction as coming from your account and going to someone else's account.

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