Doesn't the solutions of square roots break the transitive law?

Question

if the logic behind: "if a=b and b=c then c=a" is true.

why it is also true that: $\sqrt{9}=3,-3$ if $\sqrt{9}=3$ and $\sqrt{9}=-3$ but $3\neq -3$

Answer 1

We have $\sqrt 9=3$, period. However, the two solutions of $x^2=9$ are $x=+3$ and $x=-3$ (aka. $x=\pm 3=\pm\sqrt 9$).