# hno3 + caco3

To balance the equation HNO3 + CaCO3 = Ca(NO3)2 + H2O + 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.

### 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 CaCO3 = c Ca(NO3)2 + d H2O + f CO2

### Step 2: Create a System of Equations

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

```H:	1a	+	0b	=	0c	+	2d	+	0f
N:	1a	+	0b	=	2c	+	0d	+	0f
O:	3a	+	3b	=	6c	+	1d	+	2f
Ca:	0a	+	1b	=	1c	+	0d	+	0f
C:	0a	+	1b	=	0c	+	0d	+	1f

```

### Step 3: Solve For All Variables

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

• 1a - 2d = 0
• 1a - 2c = 0
• 3a + 3b - 6c - 1d - 2f = 0
• 1b - 1c = 0
• 1b - 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:

```[ 1	 0	 0	-2	 0	0]
[ 1	 0	-2	 0	 0	0]
[ 3	 3	-6	-1	-2	0]
[ 0	 1	-1	 0	 0	0]
[ 0	 1	 0	 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 = 2 (HNO3)
• b = 1 (CaCO3)
• c = 1 (Ca(NO3)2)
• d = 1 (H2O)
• f = 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.

2 HNO3 + CaCO3 = Ca(NO3)2 + H2O + CO2

Reactants Products H 2 2 ✔️ 2 2 ✔️ 9 9 ✔️ 1 1 ✔️ 1 1 ✔️

Since there is an equal number of each element in the reactants and products of 2HNO3 + CaCO3 = Ca(NO3)2 + H2O + CO2, the equation is balanced.