To balance the equation K2HPO4 + H3PO4 = KH2PO4 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 K2HPO4 + b H3PO4 = c KH2PO4

Step 2: Create a System of Equations

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

K:	2a	+	0b	=	1c
H:	1a	+	3b	=	2c
P:	1a	+	1b	=	1c
O:	4a	+	4b	=	4c

Step 3: Solve For All Variables

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

  • 2a - 1c = 0
  • 1a + 3b - 2c = 0
  • 1a + 1b - 1c = 0
  • 4a + 4b - 4c = 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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[ 2	 0	-1	0]
[ 1	 3	-2	0]
[ 1	 1	-1	0]
[ 4	 4	-4	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 (K2HPO4)
  • b = 1 (H3PO4)
  • c = 2 (KH2PO4)

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.

K2HPO4 + H3PO4 = 2 KH2PO4

Reactants Products

Since there is an equal number of each element in the reactants and products of K2HPO4 + H3PO4 = 2KH2PO4, the equation is balanced.