The formula
Each element must have the same number of atoms on both sides
What it means
Balancing an equation means solving a system: one unknown per species and one equation per element, requiring the atom count not to change. It is solved with exact fractions and then reduced to the smallest whole numbers, so the answer is the canonical balance and not one multiplied by two. The trial-and-error method taught in class reaches the same place; the difference is that this does not get stuck.
How to work it out by hand
- Put an unknown in front of each formula
- Write one equation per element: atoms on the left equal atoms on the right
- Solve the system and set one unknown to one
- Scale everything so the coefficients are the smallest whole numbers
What is worth knowing
If the equation admits more than one independent balance — which happens when there are spare species, or the reaction can be written several ways — you are told rather than having one picked in silence, because both would be equally valid and which one matters depends on the chemistry, not the algebra. And a negative coefficient means some species is on the wrong side: written that way, the reaction does not go in that direction. Redox reactions in ionic solution also need the charge balanced, which this does not do.