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QUESTION

# How do I use the algebraic method to balance the equation CaCO₃ + H₃PO₄ → Ca₃(PO₄)₂ + H₂CO₃?

The balanced equation is "3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO"_3

You write the equation using letter variables for the coefficients.

a"CaCO"_3 + b"H"_3"PO"_4 → c"Ca"_3("PO"_4)_2 + d"H"_2"CO"_3

Then you set up a series of simultaneous equations, one for each element.

"Ca": a = 3c "C": a = d "O": 3a + 4b = 8c +3d "H": 3b = 2d "P": b = 2c

Now you solve the five simultaneous equations.

It looks almost impossible, but we know that the coefficients must be integers.

Let's set c = 1

Then a = 3 and

d = a = 3

b = 2c = 2

So a = 3; b = 2; c = 1; d = 3

The balanced equation is

"3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO"_3