Fe2O3+CO=FE+CO2 balance

You want to find the coefficients a, b, c and ddsuch that

a(Fe2O3)+b(CO)→c(Fe)+d(CO2)

The idea is to make the number of each atomic species equal on the two sides of the equation equal. Then we have:

For iron: 2a=c

For oxygen: 3a+b=2d

For carbon: b=d

Oh, dear! Four unknowns but only three equations. Which implies multiple solutions. This might suggest a bad outcome. Or we could just arbitrarily set one of the variables to, say, one.

a=1. => c=2 (using the iron equation). And b=3,d=3 (using the value for a, and then solving the resulting oxygen and carbon equations).

Final result:

1(Fe2O3)+3(CO)→2(Fe)+3(CO2)