Equivalence classes in $\mathbb{Z}_n$

Question

I've the following exercise:

Solve each of the following equations in the given set $\mathbb{Z}_n$:

1) $[5]+x=[1]$ in $\mathbb{Z}_9$

2) $[2]\cdot x=[7]$ in $\mathbb{Z}_{11}$

For 1), is $x=5$ right?

For 2), is $x=7$ right?

Answer 1

The first is right as $5 + 5 =10$ is indeed congruent to $1$ modulo $9$ (the division of $10$ by $9$ leaves remainder $1$).

The second is not right as $2 \cdot 7 =14$ and $14$ is congruent to $3$ modulo $11$ and not to $7$. To help solve the second note that $18$ is congruent to $7$ modulo $11$.

Answer 2

Write equations in the congruence form $$ 5+x\equiv1\pmod9\iff x\equiv{-4}\equiv5\pmod9 $$ and for the second one $$ x\equiv 6(2x)\equiv6(7)\equiv42\equiv9\pmod{11} $$