First, formulate the half-reactions. Let's take the reaction involving Mn. Determine the oxidation number of Mn to determine the number of electrons in the reaction.
For MnO₂: x + 2(-2) = 0 --> x = +4
For MnO₄⁻: x + 4(-2) = -1 --> x = +7
Thus,
MnO₂ --> MnO₄⁻ + 3e⁻
For Cu₂: 0
For Cu²⁺: 2⁺
Thus,
Cu²⁺ + 2e⁻ --> Cu
Now, to eliminate the electrons, multiply the first reaction with 2 and the second half-reaction with 3.
2 MnO₂ --> 2 MnO₄⁻ + 6e⁻
3Cu²⁺ + 6e⁻ --> 3 Cu
Add the reactions:
2 MnO₂ + 3Cu²⁺ + 6e⁻ --> 2 MnO₄⁻ + 6e⁻ + 3 Cu
Eliminate like terms in the reactant and product side:
2 MnO₂ + 3Cu²⁺ --> 2 MnO₄⁻ + 3 Cu