# fe(no3)3 + hcl

To balance the equation Fe(NO3)3 + HCl = HNO3 + FeCl3 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 lớn represent the unknown coefficients.

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a Fe(NO3)3 + b HCl = c HNO3 + d FeCl3

### Step 2: Create a System of Equations

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

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

```

### Step 3: Solve For All Variables

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

• 1a - 1d = 0
• 3a - 1c = 0
• 9a - 3c = 0
• 1b - 1c = 0
• 1b - 3d = 0

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

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```[ 1	 0	 0	-1	0]
[ 3	 0	-1	 0	0]
[ 9	 0	-3	 0	0]
[ 0	 1	-1	 0	0]
[ 0	 1	 0	-3	0]
```

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

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

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• a = 1 (Fe(NO3)3)
• b = 3 (HCl)
• c = 3 (HNO3)
• d = 1 (FeCl3)

### 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.

Fe(NO3)3 + 3 HCl = 3 HNO3 + FeCl3

Reactants Products Fe 1 1 ✔️ 3 3 ✔️ 9 9 ✔️ 3 3 ✔️ 3 3 ✔️

Since there is an equal number of each element in the reactants and products of Fe(NO3)3 + 3HCl = 3HNO3 + FeCl3, the equation is balanced.