Under Galilean transformations between a frame A and another frame B in which A is moving with constant velocity $\mathbf V$, a velocity $\mathbf v_A$ is frame $A$ is seen as $$\mathbf v_B = \mathbf v_A + \mathbf V.$$ Use the result of part (a) [which was just a proof that forces are invariant under Galilean transformations] together with the Lorentz force law $$\mathbf F = q\mathbf E + q\mathbf v\times \mathbf B$$ to derive a Galilean transformation law for electric and magnetic fields.
Using the invariance property I see that $$\mathbf F_A = q\mathbf E_A + q\mathbf v_A\times \mathbf B_A = q\mathbf E_B + q\mathbf v_B\times \mathbf B_B = \mathbf F_B$$
Then using the definition of the Galilean tranformation I get $$\mathbf E_A + \mathbf v_A\times \mathbf B_A = \mathbf E_B + (\mathbf v_A + \mathbf V)\times \mathbf B_B = \mathbf E_B + \mathbf v_A\times \mathbf B_B + \mathbf V\times \mathbf B_B$$
Here I have no idea what to do next. I assume I'm supposed to solve for $\mathbf E_B$ and for $\mathbf B_B$ in terms of $\mathbf E_A$ and $\mathbf B_A$ but it seems like I need another equation to be able to get cancel terms or something.
Can anyone help me figure out where I'm supposed to go from here?