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##