Identify that this is a redox reaction occurring in a basic solution, involving permanganate ion (MnO4^-), bromide ion (Br^-), manganese dioxide (MnO2), and bromate ion (BrO3^-).
Write the half-reactions separately: the reduction half-reaction for MnO4^- to MnO2, and the oxidation half-reaction for Br^- to BrO3^-.
Balance each half-reaction for atoms other than O and H first, then balance oxygen atoms by adding H2O, and hydrogen atoms by adding OH^- (since the solution is basic).
Balance the charges in each half-reaction by adding electrons (e^-), ensuring that the total charge is the same on both sides of each half-reaction.
Multiply the half-reactions by appropriate coefficients so that the number of electrons lost in oxidation equals the number gained in reduction, then add the half-reactions together and simplify to get the balanced overall equation.