solving equation:

a+b+c=0

-2a-2c=1

-a-3b+2c=0

a + b  + c = 0  (1)
-2a -2c  = 1  (2)
-a - 3b  + 2c  =  0 (3)

By (2) we have a + c = -1/2 
so using this in (1) , we have -1/2 + b = 0
Or b = 1/2

Putting b = 1/2 in (3) , we have
-a -3/2 + 2c  = 0
Or -a + 2c = 3/2  (4)
Solving (4) and (2)  , we get a = -5/6 and c = 1/3 

So a = -5/6 , b = 1/2 and c = 1/3