This question is really weird and has some weird ambiguities: I'm not sure if I did it right, so please critique my logic! I also don't know how to make the whole thing a spoiler so they are kind of haphazardly placed.
We know that
- B + C = 1
- B + C + F + G = 4
- A + B + C = 7
- A + B + C + D + G = 3
- A + B + E + D + G = 2
From 1 and 3 we know that
A = 6
From 1 and 2 we know that
F + G = 3
so
G = F - 3
From 3 and 4 we know that
D + G = -4
so
D = -4 - G
so substituting the above equation for G we get
D = F - 3 - 4
so
D = F- 7
and
F = 7 + D
We can substitute the above into equation 2 and 4 to get
B + C + (7 + D) + (3 - G) = 4
so
D = -7
and
A + B + C + (-7) + G = 3
so
G = 3
and substituting this back into equation 2 gives us
F = 0
This is where things get weird:
We know from equations 4 and 5 that
C - E = 1
and using that in equation 1 we find that
C = -B + 1
therefore
B = -E
This means that we can use literally any value for B, and as long as C and E follow the above conventions, all the equations will still be true.