# bahco3 na2co3

To balance the equation Ba(HCO3)2 + Na2CO3 = BaCO3 + NaHCO3 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 lớn represent the unknown coefficients.

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a Ba(HCO3)2 + b Na2CO3 = c BaCO3 + d NaHCO3

### Step 2: Create a System of Equations

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

```Ba:	1a	+	0b	=	1c	+	0d
H:	2a	+	0b	=	0c	+	1d
C:	2a	+	1b	=	1c	+	1d
O:	6a	+	3b	=	3c	+	3d
Na:	0a	+	2b	=	0c	+	1d

```

### Step 3: Solve For All Variables

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

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

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

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

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

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

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• a = 1 (Ba(HCO3)2)
• b = 1 (Na2CO3)
• c = 1 (BaCO3)
• d = 2 (NaHCO3)

### 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.

Ba(HCO3)2 + Na2CO3 = BaCO3 + 2 NaHCO3

Reactants Products Ba 1 1 ✔️ 2 2 ✔️ 3 3 ✔️ 9 9 ✔️ 2 2 ✔️

Since there is an equal number of each element in the reactants and products of Ba(HCO3)2 + Na2CO3 = BaCO3 + 2NaHCO3, the equation is balanced.