To balance the equation Fe3O4 + H2 = Fe + H2O 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 Fe3O4 + b H2 = c Fe + d H2O

Step 2: Create a System of Equations

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

Fe:	3a	+	0b	=	1c	+	0d
O:	4a	+	0b	=	0c	+	1d
H:	0a	+	2b	=	0c	+	2d

Step 3: Solve For All Variables

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

  • 3a - 1c = 0
  • 4a - 1d = 0
  • 2b - 2d = 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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[ 3	 0	-1	 0	0]
[ 4	 0	 0	-1	0]
[ 0	 2	 0	-2	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 = 1 (Fe3O4)
  • b = 4 (H2)
  • c = 3 (Fe)
  • d = 4 (H2O)

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.

Fe3O4 + 4 H2 = 3 Fe + 4 H2O

Reactants Products

Since there is an equal number of each element in the reactants and products of Fe3O4 + 4H2 = 3Fe + 4H2O, the equation is balanced.