To balance the equation CH3CHO + Br2 + H2O = CH3COOH + HBr 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 CH_{3}CHO + b Br_{2} + c H_{2}O = d CH_{3}COOH + f HBr

### Step 2: Create a System of Equations

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

**C**: 2a + 0b + 0c = 2d + 0f
**H**: 4a + 0b + 2c = 4d + 1f
**O**: 1a + 0b + 1c = 2d + 0f
**Br**: 0a + 2b + 0c = 0d + 1f

### Step 3: Solve For All Variables

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

- 2a - 2d = 0
- 4a + 2c - 4d - 1f = 0
- 1a + 1c - 2d = 0
- 2b - 1f = 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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[ 2 0 0 -2 0 0] [ 4 0 2 -4 -1 0] [ 1 0 1 -2 0 0] [ 0 2 0 0 -1 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 (CH3CHO)
- b = 1 (Br2)
- c = 1 (H2O)
- d = 1 (CH3COOH)
- f = 2 (HBr)

### 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.

CH_{3}CHO + Br_{2} + H_{2}O = CH_{3}COOH + 2 HBr

C | 2 | 2 | ✔️ |
---|---|---|---|

H | 6 | 6 | ✔️ |

O | 2 | 2 | ✔️ |

Br | 2 | 2 | ✔️ |

Since there is an equal number of each element in the reactants and products of CH3CHO + Br2 + H2O = CH3COOH + 2HBr, the equation is balanced.

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