# alcl3 alno33

To balance the equation HNO3 + AlCl3 = Al(NO3)3 + HCl 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 HNO3 + b AlCl3 = c Al(NO3)3 + d HCl

### Step 2: Create a System of Equations

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

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

```

### Step 3: Solve For All Variables

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

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

• a = 3 (HNO3)
• b = 1 (AlCl3)
• c = 1 (Al(NO3)3)
• d = 3 (HCl)

### 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.

3 HNO3 + AlCl3 = Al(NO3)3 + 3 HCl

Reactants Products H 3 3 ✔️ 3 3 ✔️ 9 9 ✔️ 1 1 ✔️ 3 3 ✔️

Since there is an equal number of each element in the reactants and products of 3HNO3 + AlCl3 = Al(NO3)3 + 3HCl, the equation is balanced.