What is a balanced chemical equation for iron oxide reacting with carbon monoxide that produces atomic iron and carbon dioxide?

Hello! Most likely, it is iron (III) oxide, Fe_2 O_3. Carbon monoxide is identified as CO, and carbon dioxide is identified asCO_2.
The unbalanced reaction is Fe_2 O_3 + CO -> Fe + CO_2. To balance it, place undetermined coefficients before each molecule and find them:
a_1 * Fe_2 O_3 + a_2 * CO = a_3 * Fe + a_4 * CO_2.
The quantities of each atom on both sides must be the same. There are three types of atoms in this reaction: Fe, C, and O. For Fe atoms, we have 2a_1 = a_3, for C atoms, we have a_2 = a_4, and for O atoms we have 3a_1 + a_2 = 2a_4.
Substitute a_4 = a_2 from the second equation to the third and obtain 3a_1 + a_2 = 2a_2 or a_1 = 1/3 a_2. Substitute this into the first equation and obtain 2/3 a_2 = a_3.
All coefficients must be natural numbers, so the smallest value for a_2 is 3. Then, we get a_3 = 2/3 a_2 = 2, a_1 = 1/3 a_2 = 1, and a_4 = a_2 = 3.
The balanced reaction is as follows:
Fe_2 O_3 + 3 CO -> 2 Fe + 3 CO_2.
http://www.chm.bris.ac.uk/motm/co/coh.htm