Each public key is compatible with exactly one private key, in the sense that a server or any other computer can use a public key to decrypt a message that was encrypted using the corresponding private key. However, knowing the public key does not allow the server to find the private key.1 The public and private keys are generated at the same time by
ssh-keygen or whatever equivalent tool you use to make your keys.
As part of (a simplified "toy" version of) the SSH login process, the server makes up a string, called a nonce, and sends it to the client (your personal computer). The client encrypts the nonce with your private key and sends the result back to the server. Then the server tries to decrypt that result with each public key listed in the
.ssh/authorized_keys file, and if it finds one for which the decryption process results in the original nonce, it grants you access. Note that at no point in this process does the server ever have access to, or know anything about, the private key.
Since each public key corresponds to exactly one private key, in a manner fixed by the key's encryption algorithm (RSA or DSA or ECDSA), you cannot change what private key the server will accept for a given public key. If you want to use a different private key, you need to generate the corresponding public key and put it in the
.ssh/authorized_keys file as well.
1technically it could, but it would take longer than the age of the universe