# hno3 ra cu no3 2

To balance the equation CuNO3 + HNO3 = Cu(NO3)2 + NO + 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 tướng represent the unknown coefficients.

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a CuNO3 + b HNO3 = c Cu(NO3)2 + d NO + f H2O

### Step 2: Create a System of Equations

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

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

```

### Step 3: Solve For All Variables

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

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

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

```[ 1	 0	-1	 0	 0	0]
[ 1	 1	-2	-1	 0	0]
[ 3	 3	-6	-1	-1	0]
[ 0	 1	 0	 0	-2	0]
```

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

Simplify the result to tướng get the lowest, whole integer values.

• a = 3 (CuNO3)
• b = 4 (HNO3)
• c = 3 (Cu(NO3)2)
• d = 1 (NO)
• f = 2 (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.

3 CuNO3 + 4 HNO3 = 3 Cu(NO3)2 + NO + 2 H2O

Reactants Products Cu 3 3 ✔️ 7 7 ✔️ 21 21 ✔️ 4 4 ✔️

Since there is an equal number of each element in the reactants and products of 3CuNO3 + 4HNO3 = 3Cu(NO3)2 + NO + 2H2O, the equation is balanced.