To balance the equation C2H4(OH)2 + Cu(OH)2 = C2H4O2Cu + 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 tướng represent the unknown coefficients.

a C_{2}H_{4}(OH)_{2} + b Cu(OH)_{2} = c C_{2}H_{4}O_{2}Cu + d H_{2}O

### Step 2: Create a System of Equations

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

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

### Step 3: Solve For All Variables

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

- 2a - 2c = 0
- 6a + 2b - 4c - 2d = 0
- 2a + 2b - 2c - 1d = 0
- 1b - 1c = 0

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

[ 2 0 -2 0 0] [ 6 2 -4 -2 0] [ 2 2 -2 -1 0] [ 0 1 -1 0 0]

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

Simplify the result to tướng get the lowest, whole integer values.

- a = 1 (C2H4(OH)2)
- b = 1 (Cu(OH)2)
- c = 1 (C2H4O2Cu)
- d = 2 (H2O)

### Step 4: Substitute Coefficients and Verify Result

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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.

C_{2}H_{4}(OH)_{2} + Cu(OH)_{2} = C_{2}H_{4}O_{2}Cu + 2 H_{2}O

C | 2 | 2 | ✔️ |
---|---|---|---|

H | 8 | 8 | ✔️ |

O | 4 | 4 | ✔️ |

Cu | 1 | 1 | ✔️ |

Since there is an equal number of each element in the reactants and products of C2H4(OH)2 + Cu(OH)2 = C2H4O2Cu + 2H2O, the equation is balanced.

The law of conservation of mass states that matter cannot be created or destroyed, which means there must be the same number atoms at the kết thúc of a chemical reaction as at the beginning. To be balanced, every element in C2H4(OH)2 + Cu(OH)2 = C2H4O2Cu + H2O must have the same number of atoms on each side of the equation. When using the inspection method (also known as the trial-and-error method), this principle is used to tướng balance one element at a time until both sides are equal and the chemical equation is balanced.

### Step 1: Count the number of each element on the left and right hand sides

Reactants (Left Hand Side) | Products (Right Hand Side) | ||||||
---|---|---|---|---|---|---|---|

Reactants | Products | ||||||

C_{2}H_{4}(OH)_{2} | Cu(OH)_{2} | Total | C_{2}H_{4}O_{2}Cu | H_{2}O | Total | ||

C | 2 | 2 | 2 | 2 | ✔️ | ||

H | 6 | 2 | 8 | 4 | 2 | 6 | ❌ |

O | 2 | 2 | 4 | 2 | 1 | 3 | ❌ |

Cu | 1 | 1 | 1 | 1 | ✔️ |

### Step 2: Multiply coefficients for compounds to tướng balance out each element

For each element that is not equal, try to tướng balance it by adding more of it to tướng the side with less. Sometimes there may be multiple compounds with that element on one side, ví you'll need to tướng use your best judgement and be prepared to tướng go back and try the other options.

**H**is not balanced. Add**1**molecule of**H**to tướng the product (right-hand) side to tướng balance Hydrogen:_{2}O

C_{2}H_{4}(OH)_{2}+ Cu(OH)_{2}= C_{2}H_{4}O_{2}Cu + 2H_{2}OReactants Products C 2 2 ✔️ H 8 8 ✔️ O 4 4 ✔️ Cu 1 1 ✔️

### Step 3: Verify that the equation is balanced

Since there are an equal number of atoms of each element on both sides, the equation is balanced.

C_{2}H_{4}(OH)_{2} + Cu(OH)_{2} = C_{2}H_{4}O_{2}Cu + 2H_{2}O

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