I'm trying to follow this example on solving systems of congruences to understand CRT, however I do not understand how the author solves the following congruence:
$$ 7j + 6 \equiv 4 \pmod 5 \\ j \equiv 4 \pmod 5 $$
If I were to attempt to solve this congruence, I would try $$ 7j + 6 \equiv 4 \pmod 5 \\ 7j \equiv -2 \pmod 5 $$ but I do not know if subtracting 6 from both sides is valid.
Assuming it is, I would just guess and check numbers until I found a $j$ such that $5|(7j + 2)$.
So how do I solve this congruence (ideally, without guessing and checking)?
We have $$7j+6\equiv 4\bmod 5,$$ subtracting $6$ gives $$7j\equiv -2\equiv 3\bmod 5,$$ that is $$2j\equiv3\bmod 5.$$ Now we must find a number $b$ such that $2b\equiv 1\bmod 5$ in order to isolate $j$. It can be checked that $b=3$ does the trick. So, multiplying the congruence through by $3$ gives $$j\equiv 9\equiv 4\bmod 5.$$