There is no EASY way to solve this set manually.
A hand held calculator would invert that matrix "instantly" and doing it by hand is simply onerous.
Looking at the coefficients, I think I would eliminate the z variable first.
multiply the 3rd equation by 2 and add to the first equation, then multiply the 2nd equation by 2 and the 3rd equation by 3 and add. That will eliminate the z and leave you with 2 equations in x and y. Then any convenient method will get you x and y.
If you are still stuck, make a comment so that I can try to help further.