elementary number theory - Find all $x$ such that $11mid 3x+7$

I found this question in Beachy and Blair: Abstract algebra book, they even have a solution to this but its not satisfactory for me. They only say "$x\equiv 5 \pmod{11}$ ". Which one can "feel" simply by trial and error. I would like to know what is the proper approach. Thank you in advance!

Answer

We need $3x+7\equiv 0\pmod{11}$

Add 4 to both sides:

$$3x+11\equiv 4\pmod{11}$$

reduce:

$$3x \equiv 4\pmod{11}$$

multiply both sides by a number to make the coefficient on the left equivalent to $1$. In this case, $4$ works:

$$12x\equiv 16\pmod{11}$$

reduce:

$$x\equiv 5\pmod{11}$$

Does that work for you?