Can someone please help me out with this modular arithmetic problem?

Question

I've been trying to see how to calculate this math problem: 10+9a{ mod } 23 =14.I know that a=3, but I don't know the steps to apply to these digits to get 'a' if I were to calculate it. 10+9*3{ mod }23=14. I'm just looking for the steps on how to find 'a'. If I were asked to calculate it. Thanks so much in advance. And Please I would understand more if you could implement the answer mathematically like the way it's in the question above

Answer 1

$10+9a\equiv14\bmod23\implies9a\equiv4\bmod23\implies9a\equiv27\bmod23\implies a\equiv3\bmod23.$

Answer 2

It implies $9a \pmod{23}=4$ and as $gcd(9,23)=1$ it guarantees the existence of one $a$ satisfying this (To see why, suppose not, and $9a=9b \pmod{23}$ for some distinct $a,b$ such that $|a-b|<23$. As $gcd(9,23)=1$ you can cancel the $9$ both sides to obtain $|a-b| = 0 \pmod {23}$ which is false as $|a-b|<23$, thus one should have $a=b$). Now use brute force. (I am not sure if other methods exist, which is unlikely as they rely on computation, it is fine if you show the existence of such $a$, then you obtain infinitely many $a$)