Good clear explanation Rob.
If you'll allow me to add that balancing equations comes from the law of conservation of matter, which says that matter cannot be created nor destroyed in chemical reactions (putting E=mc2 to one side for a moment)
Consequently the particles (atoms) that you begin with in a chemical reaction on the left hand side must appear in the same quantities on the right hand side in the products.
It then becomes a matter of applying simple mathematics to the compounds containing the atoms - remembering that each substance has a formula that cannot be interfered with - balancing numbers must be placed in front of the formulae and multiply ALL of the atoms in the formulae.
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