To balance the equation Al + Fe2O3 = Al2O3 + Fe using the algebraic method step-by-step, you must have experience solving systems of linear equations. The most common methods are substitution/elimination and linear algebra, but any similar method will work.

Step 1: Label Each Compound With a Variable

Label each compound (reactant or product) in the equation with a variable to tát represent the unknown coefficients.

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a Al + b Fe2O3 = c Al2O3 + d Fe

Step 2: Create a System of Equations

Create an equation for each element (Al, Fe, O) where each term represents the number of atoms of the element in each reactant or product.

Al:	1a	+	0b	=	2c	+	0d
Fe:	0a	+	2b	=	0c	+	1d
O:	0a	+	3b	=	3c	+	0d

Step 3: Solve For All Variables

Use substitution, Gaussian elimination, or a calculator to tát solve for each variable.

  • 1a - 2c = 0
  • 2b - 1d = 0
  • 3b - 3c = 0

Use your graphing calculator's rref() function (or an online rref calculator) to tát convert the following matrix into reduced row-echelon-form:

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[ 1	 0	-2	 0	0]
[ 0	 2	 0	-1	0]
[ 0	 3	-3	 0	0]

The resulting matrix can be used to tát determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.

Simplify the result to tát get the lowest, whole integer values.

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  • a = 2 (Al)
  • b = 1 (Fe2O3)
  • c = 1 (Al2O3)
  • d = 2 (Fe)

Step 4: Substitute Coefficients and Verify Result

Count the number of atoms of each element on each side of the equation and verify that all elements and electrons (if there are charges/ions) are balanced.

2 Al + Fe2O3 = Al2O3 + 2 Fe

Reactants Products

Since there is an equal number of each element in the reactants and products of 2Al + Fe2O3 = Al2O3 + 2Fe, the equation is balanced.