You can construct examples of reactants and products containing the same elements for which no set of coefficients can be found conserving atoms of each element. Some chemical equations can be balanced by several different sets of coefficients.
An example of each type is shown on the screen. Try to balance the first equation, and in the second example verify that all the different sets of coefficients do indeed each balance the equation.
Sometimes you can't find an element that appears in only one reactant and one product. These equations may be balanced by the more advanced techniques described in the article by Blakley.