# p2o5 + h2o

To balance the equation P2O5 + H2O = H3PO4 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.

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### 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.

a P2O5 + b H2O = c H3PO4

### Step 2: Create a System of Equations

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

P: 2a + 0b = 1c O: 5a + 1b = 4c H: 0a + 2b = 3c

### Step 3: Solve For All Variables

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

• 2a - 1c = 0
• 5a + 1b - 4c = 0
• 2b - 3c = 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] [ 5 1 -4 0] [ 0 2 -3 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.

• a = 1 (P2O5)
• b = 3 (H2O)
• c = 2 (H3PO4)

### 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.

P2O5 + 3 H2O = 2 H3PO4

Reactants Products P 2 2 ✔️ 8 8 ✔️ 6 6 ✔️

Since there is an equal number of each element in the reactants and products of P2O5 + 3H2O = 2H3PO4, the equation is balanced.

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