Why don't you just solve the matrix on your calc, or with row reduction, then add a+b+c?
Or to get it algebraically, call the equations P, Q and R respectively, so
P: 4a + 3b + c = 27
Q: 3a + b + 4c = 16
R: 9a + 2b + 3c = 33
Solve these for b first, since the multiples are easy:
R-2Q = 3a - 5c = 1
P-3Q = -5a - 11c = -21
5(R-2Q) = 15a - 25c = 5
3(P-3Q) = -15a - 33c = -63
Add these together and you get -58c = 58, so c=1, therefore a=2. Feed a and c into any of the original equations and you get b=6. So a+b+c=9, and twice this is 18.
Correction: -58c=-58, so c=1.