# hno3 ra al(no3)3

To balance the equation Al + HNO3 = Al(NO3)3 + N2 + 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.

### 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.

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a Al + b HNO3 = c Al(NO3)3 + d N2 + f H2O

### Step 2: Create a System of Equations

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

```Al:	1a	+	0b	=	1c	+	0d	+	0f
H:	0a	+	1b	=	0c	+	0d	+	2f
N:	0a	+	1b	=	3c	+	2d	+	0f
O:	0a	+	3b	=	9c	+	0d	+	1f

```

### Step 3: Solve For All Variables

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

• 1a - 1c = 0
• 1b - 2f = 0
• 1b - 3c - 2d = 0
• 3b - 9c - 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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```[ 1	 0	-1	 0	 0	0]
[ 0	 1	 0	 0	-2	0]
[ 0	 1	-3	-2	 0	0]
[ 0	 3	-9	 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.

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• a = 10 (Al)
• b = 36 (HNO3)
• c = 10 (Al(NO3)3)
• d = 3 (N2)
• f = 18 (H2O)

### 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.

10 Al + 36 HNO3 = 10 Al(NO3)3 + 3 N2 + 18 H2O

Reactants Products Al 10 10 ✔️ 36 36 ✔️ 36 36 ✔️ 108 108 ✔️

Since there is an equal number of each element in the reactants and products of 10Al + 36HNO3 = 10Al(NO3)3 + 3N2 + 18H2O, the equation is balanced.