This is my first math problem set, and I am a little confused. This is my solution to the problem, however it think it might be a little too easy...
I have a Field $F$ and $ a, b \in F$. I am supposed to show that the equation $ a + x = b $ has a unique solution.
First I show existence:
$ a + x = b \Leftrightarrow a + x + (-a) = b + (-a) \Leftrightarrow a + (-a) + x = b + (-a) \Leftrightarrow 0 + x = b - a \Leftrightarrow x = b - a. $
Now I want to show that the solution is indeed unique: Suppose there is another number $y$ for which $y = b - a$. Then $ y = b - a = x \Rightarrow y = x$
But again, I am really not sure about the last part. Can you tell me if I am wrong, and if so, give me a hint as to why this wouldn't work?