$3x\equiv7\pmod{11}, 5y\equiv9\pmod{11}$. Find the number which $x+y\pmod{11}$ is congruent to. Given that $3x\equiv7\pmod{11}, 5y\equiv9\pmod{11}$. Find the number which $x+y\pmod{11}$ is congruent to. I'm thinking $20\equiv9\pmod{11}$, But I am having trouble find a number $3x$ that  is divisible by $3$? Is there a better way of solving this problem.

By inspection, $x = 6, y = 4$ is a solution.
For a more general approach, try solving $$3x \equiv 1 \bmod{11}$$ and $$5y \equiv 1 \bmod{11}$$ using Extended GCD, and multiply the first by 7 and the second by 9. To get more solutions, just increase/decrease either $x$ or $y$ by 11. 
(However, since you're just interested in what $x + y$ is mod 11, you can skip the "get more solutions" part; they would all get the same answer.)
