I think you must have mistyped the equation 

1+3c+2d+2f=110

as none of the others had a number in front
that you had to subtract from both sides, so
if the answer I get is not the right one it's
because you mistyped that 1 in front of the 
fourth equation

a+b+c+d+e+f=0
a-2b=0
a+2b+3d+2e=220
1+3c+2d+2f=110
a+2b+4d+e+5f=240
a+2d+3e+4f=165 

1a + 1b + 1c + 1d + 1e + 1f =   0
1a - 2b + 0c + 0d + 0e + 0f =   0
1a + 2b + 0c + 3d + 2e + 0f = 220
0a + 0b + 3c + 2d + 0e + 2f = 110-1
1a + 2b + 0c + 4d + 1e + 5f = 240
1a + 0b + 0c + 2d + 3e + 4f = 165

Simplifying the equation that I think
you mistyped:

1a + 1b + 1c + 1d + 1e + 1f =   0
1a - 2b + 0c + 0d + 0e + 0f =   0
1a + 2b + 0c + 3d + 2e + 0f = 220
0a + 0b + 3c + 2d + 0e + 2f = 109
1a + 2b + 0c + 4d + 1e + 5f = 240
1a + 0b + 0c + 2d + 3e + 4f = 165 

This is the matrix AX=B form:

 

The inverse of that matrix is:



We multiply both sides of the  equation by that
inverse matrix, getting it in the form 

 

Matrix multiplication. though not commutative, is
associative, so we move the parentheses and change the
 form to the
 form



Next we do the matrix multiplication on both sides, and
since  we have the  form:



When we make the multiplication on the left, we
have the final solution, the  form:



The answer you wanted is probably all different numbers since you 
probably mistyped that 1 in front of the 4th equation instead of
what it should have been.

I did the above on a TI-84 calculator.

Edwin