Suppose there are B boys and G girls.

Your brother has B-1 brothers.  [He is not his own brother]
Your brother has G sisters.

B-1 = G

Your sister has B brothers
Your sister has G-1 sisters. [She is not her own sister]

B = 2(G-1)

So the system of equations is

B-1 = G
B = 2(G-1)

Substitute 2(G-1) for B in the first equation:

2(G-1)-1 = G
  2G-2-1 = G
    2G-3 = G
       G = 3

Substitute 3 for G in

B-1 = G
B-1 = 3
  B = 4

There are 4 boys and 3 girls.

Edwin