To balance the equation Cu + CH3COOH = Cu(CH3COO)3 + H2 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 Cu + b CH_{3}COOH = c Cu(CH_{3}COO)_{3} + d H_{2}

### Step 2: Create a System of Equations

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

**Cu**: 1a + 0b = 1c + 0d
**C**: 0a + 2b = 6c + 0d
**H**: 0a + 4b = 9c + 2d
**O**: 0a + 2b = 6c + 0d

### Step 3: Solve For All Variables

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

- 1a - 1c = 0
- 2b - 6c = 0
- 4b - 9c - 2d = 0
- 2b - 6c = 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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[ 1 0 -1 0 0] [ 0 2 -6 0 0] [ 0 4 -9 -2 0] [ 0 2 -6 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 = 2 (Cu)
- b = 6 (CH3COOH)
- c = 2 (Cu(CH3COO)3)
- d = 3 (H2)

### 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 Cu + 6 CH_{3}COOH = 2 Cu(CH_{3}COO)_{3} + 3 H_{2}

Cu | 2 | 2 | ✔️ |
---|---|---|---|

C | 12 | 12 | ✔️ |

H | 24 | 24 | ✔️ |

O | 12 | 12 | ✔️ |

Since there is an equal number of each element in the reactants and products of 2Cu + 6CH3COOH = 2Cu(CH3COO)3 + 3H2, the equation is balanced.

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