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The reaction is impossible to balance by manipulating coefficients I feel you over-interpreted the statement. Instead of a pure mathematical reasoning, think in terms of mass balance. New atoms cannot be created or destroyed during a chemical reaction. Let us see the classical $\ce{Fe^{2+}}$ and $\ce{MnO4-}$ reaction. The case of $\ce{Fe^{2+}->Fe^{3+} + ...

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