You can make the arithmetic easier to handle by simplifying some of the equations.
For example your equation 1 could have been divided throughout by 2.
With the two equations you are left with, the first can be divided throughout by 3 and the second by 2.
Also, the calculation becomes easier with a better choice of which equations to combine.
For example you could have started by adding eq. 2 to eq. 1 to get you 16a + 14b - 13c = 0, (with the general idea of reducing the size of the coefficients), and then adding twice this to eq. 2.
That gets you 74b - 37c = 0, and that can be divided throughout by 37, yielding 2b - c = 0.
There are other methods, but when dealing with integers like this, this is as good as any.