6times 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

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.

He secknd step was the answer to them