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.

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### 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.

a Fe_{3}O_{4} + b H_{2} = c Fe + d H_{2}O

### 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.

- 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.

Fe_{3}O_{4} + 4 H_{2} = 3 Fe + 4 H_{2}O

Fe | 3 | 3 | ✔️ |
---|---|---|---|

O | 4 | 4 | ✔️ |

H | 8 | 8 | ✔️ |

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

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