Suppose I was trying to give some $x \in \mathbb{Z}$ such that $3x \equiv 5 \mod 7$ then multiplying by $12$ gives $36x \equiv 60 \mod 7 \implies x \equiv 4 \mod 7$ so we can easily deduce solutions from this.
Then my question is why exactly does this not work?
$3x \equiv 5 \mod 7$ then multiplying by $2$ gives $6x \equiv 10 \mod 7 \implies -x \equiv 10 \mod 7 \implies x \equiv -10 \mod 7 \implies x \equiv 3 \mod 7$
But obviously these solutions do not match?