To balance the equation (NH4)3PO4 = H3PO4 + NH3 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 lớn represent the unknown coefficients.

a (NH_{4})_{3}PO_{4} = b H_{3}PO_{4} + c NH_{3}

### Step 2: Create a System of Equations

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

**N**: 3a = 0b + 1c
**H**: 12a = 3b + 3c
**P**: 1a = 1b + 0c
**O**: 4a = 4b + 0c

### Step 3: Solve For All Variables

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

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

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

Xem thêm: sio2 + c

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

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

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

- a = 1 ((NH4)3PO4)
- b = 1 (H3PO4)
- c = 3 (NH3)

### 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.

(NH_{4})_{3}PO_{4} = H_{3}PO_{4} + 3 NH_{3}

N | 3 | 3 | ✔️ |
---|---|---|---|

H | 12 | 12 | ✔️ |

P | 1 | 1 | ✔️ |

O | 4 | 4 | ✔️ |

Since there is an equal number of each element in the reactants and products of (NH4)3PO4 = H3PO4 + 3NH3, the equation is balanced.

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