Given that $-6x \equiv -8 \pmod{7}$, show that $x \equiv 6 \pmod{7}$

Since $8 \equiv 1 \pmod 7$, we can rewrite the text as \[ 6x \equiv 1 \pmod 7. \] Multiplying by $6$, we get \[ 36x \equiv 6 \pmod 7, \] and since $36 \equiv 1 \pmod 7$, we finally get \[ x \equiv 6 \pmod 7, \] as desired.

Edit: even faster, $-6 \equiv 1 \pmod 7$, and $-8 \equiv 6 \pmod 7$, so the thesis follows immediately.