How to calculate this congruency? Let's say I have this linear congruency: $2x + 1234 = 7 \mod 17$. Without "$+1234$" I would've used the following formulas: $x = x_0 + k(\dfrac{m}{gcd(a, m)})$, whereas $ax_0 + my_0 = b$. But I don't know what to do with $1234$? Thank you in advance.

I would calculate everything mod. $17$: $\;1234\equiv 10\mod 17$, so the equation becomes $$2x+10\equiv 7\mod17\iff 2x\equiv-3\equiv14\mod 17. $$ Now as $2$ is a unit mod. $17$, we may apply the _cancellation law_ : $$2x\equiv 14=2\cdot 7\mod17\iff x\equiv 7\mod 17.$$