Question

6 times a larger number (so 6y) is equal (=) to the sum of a snaller number and 18 (x+18)

6y = x+18

Twice of larger number (2y) and the difference of smaller number (x) is equal to 6

2y - x = 6

Answer 1

I'm not sure what techniques you've been looking at to solve a system like this, but substitution would be a good one to use. Solve one of the equations for one of the variables (pick an easy one). How about using the first equation and solving for x:
6y = x + 18
x = 6y - 18
Now, plug in the expression for x into the other equation:
2y - (6y - 18) = 6
Solve that for y, then use the answer to find x.

Answer 2

6y = x + 18
2y = x + 6 subtract :-

4y = 12
y = 3

substitute y=3 into second equation:-
6 = x + 6
so x = 0

solution is x=0.y=3