# na3po4 + cacl2

To balance the equation Na3PO4 + CaCl2 = NaCl + Ca3(PO4)2 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 Na3PO4 + b CaCl2 = c NaCl + d Ca3(PO4)2

### Step 2: Create a System of Equations

Create an equation for each element (Na, P.., O, Ca, Cl) where each term represents the number of atoms of the element in each reactant or product.

```Na:	3a	+	0b	=	1c	+	0d
P:	1a	+	0b	=	0c	+	2d
O:	4a	+	0b	=	0c	+	8d
Ca:	0a	+	1b	=	0c	+	3d
Cl:	0a	+	2b	=	1c	+	0d

```

### Step 3: Solve For All Variables

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

• 3a - 1c = 0
• 1a - 2d = 0
• 4a - 8d = 0
• 1b - 3d = 0
• 2b - 1c = 0

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

```[ 3	 0	-1	 0	0]
[ 1	 0	 0	-2	0]
[ 4	 0	 0	-8	0]
[ 0	 1	 0	-3	0]
[ 0	 2	-1	 0	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 = 2 (Na3PO4)
• b = 3 (CaCl2)
• c = 6 (NaCl)
• d = 1 (Ca3(PO4)2)

### 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 Na3PO4 + 3 CaCl2 = 6 NaCl + Ca3(PO4)2

Reactants Products Na 6 6 ✔️ 2 2 ✔️ 8 8 ✔️ 3 3 ✔️ 6 6 ✔️

Since there is an equal number of each element in the reactants and products of 2Na3PO4 + 3CaCl2 = 6NaCl + Ca3(PO4)2, the equation is balanced.