It's in fact a little more advanced than you imagine it. (So expect this answer to be a bit more in-depth.)
There is no such thing as an accounts "balance". It only exists implicitly.
When people make transactions, they actually create outputs for a certain amount of bitcoins. Using a special script language, the person making the transaction can specify the requirement for spending that output. There's a whole range of options you have to specify this.
The most common output script is something that holds the following condition "to spend this transaction, the spender must sign this hash with the private key from this address". So this basically means that the owner of that address can spend it. This are the type of outputs that you will probably generate when you "send money to an address". They are called pay-to-pubkey-hash transactions.
Other possible output scripts can be stupid things like "anyone can spend this" to very complex things like "at least 3 of the private keys of the following 5 addresses must sign this hash", which is a so-called m-of-n transaction.
This brings us back to balances. When your wallet app says you have a balance of X btc, it just searches all unspent outputs that you are able to spend with your private keys; this are all pay-to-pubkey-hash transactions with one of your addresses in them.
Now, when you want to spend some of the money from your "balance" and send it to an address, you will need to make a transaction. The way transactions are created is as follows:
- Take a number of unspent outputs from previous transactions and use them as inputs.
- Make a new output of which the output amount is equal or less than the sum of all input values. Your wallet will most probably make this output a pay-to-pubkey-hash output to the address you wanted to send money to.
- Because probably the inputs will not exactly match, you will add another output so that the sum of all new outputs is equal to the sum of the inputs. This output will also be a pay-to-pubkey-hash output to a new change address or to the same address as you are spending from (different wallet apps do this differently).
- Note that if the total output amount is less than the total input amount, the difference is considered the transaction fee and this will go to the miner of the block your transaction will appear in.
- Then lastly the wallet app has to prove it has the right to spend the inputs you used in the transaction. It can do this to provide an input script (technically it's called a scriptPubKey). For the common pay-to-pubkey-hash, this script will contain your public key and the signature you made with your private key. This input script is used by miners to verify if you have the right to make this transaction.
- The transaction is then broadcast to the Bitcoin network.
After this transaction has been received by other clients and is verified by a miner, all clients will remove the outputs used in the transaction from their collection of "unspent outputs". So if you would try to spend it again, that won't be possible because miners will not accept it because the output you want to spend does not exist.
I hope you can understand that using this methodology, it is not possible to do a thing like "subtract the amount from person A's balance and add it to person B's balance". When making transactions, you basically destroy some outputs to generate new ones.