To balance the equation (NH2)2CO + H2O = NH3 + CO2 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 đồ sộ represent the unknown coefficients.

a (NH_{2})_{2}CO + b H_{2}O = c NH_{3} + d CO_{2}

### Step 2: Create a System of Equations

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

**N**: 2a + 0b = 1c + 0d
**H**: 4a + 2b = 3c + 0d
**C**: 1a + 0b = 0c + 1d
**O**: 1a + 1b = 0c + 2d

### Step 3: Solve For All Variables

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

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

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

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

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

Simplify the result đồ sộ get the lowest, whole integer values.

- a = 1 ((NH2)2CO)
- b = 1 (H2O)
- c = 2 (NH3)
- d = 1 (CO2)

### 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_{2})_{2}CO + H_{2}O = 2 NH_{3} + CO_{2}

N | 2 | 2 | ✔️ |
---|---|---|---|

H | 6 | 6 | ✔️ |

C | 1 | 1 | ✔️ |

O | 2 | 2 | ✔️ |

Since there is an equal number of each element in the reactants and products of (NH2)2CO + H2O = 2NH3 + CO2, the equation is balanced.

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