To balance the equation Al2(SO4)3 + HCl = AlCl3 + H2SO4 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 lớn represent the unknown coefficients.

a Al_{2}(SO_{4})_{3} + b HCl = c AlCl_{3} + d H_{2}SO_{4}

### Step 2: Create a System of Equations

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

**Al**: 2a + 0b = 1c + 0d
**S**: 3a + 0b = 0c + 1d
**O**: 12a + 0b = 0c + 4d
**H**: 0a + 1b = 0c + 2d
**Cl**: 0a + 1b = 3c + 0d

### Step 3: Solve For All Variables

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

- 2a - 1c = 0
- 3a - 1d = 0
- 12a - 4d = 0
- 1b - 2d = 0
- 1b - 3c = 0

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

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[ 2 0 -1 0 0] [ 3 0 0 -1 0] [ 12 0 0 -4 0] [ 0 1 0 -2 0] [ 0 1 -3 0 0]

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

Simplify the result lớn get the lowest, whole integer values.

- a = 1 (Al2(SO4)3)
- b = 6 (HCl)
- c = 2 (AlCl3)
- d = 3 (H2SO4)

### 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.

Al_{2}(SO_{4})_{3} + 6 HCl = 2 AlCl_{3} + 3 H_{2}SO_{4}

Al | 2 | 2 | ✔️ |
---|---|---|---|

S | 3 | 3 | ✔️ |

O | 12 | 12 | ✔️ |

H | 6 | 6 | ✔️ |

Cl | 6 | 6 | ✔️ |

Since there is an equal number of each element in the reactants and products of Al2(SO4)3 + 6HCl = 2AlCl3 + 3H2SO4, the equation is balanced.

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