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

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

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.