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

Sep 16, 2014

The balanced equation is ${\text{3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO}}_{3}$

#### Explanation:

You write the equation using letter variables for the coefficients.

$a {\text{CaCO"_3 + b"H"_3"PO"_4 → c"Ca"_3("PO"_4)_2 + d"H"_2"CO}}_{3}$

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

$\text{Ca} : a = 3 c$
$\text{C} : a = d$
$\text{O} : 3 a + 4 b = 8 c + 3 d$
$\text{H} : 3 b = 2 d$
$\text{P} : b = 2 c$

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

$d = a = 3$

$b = 2 c = 2$

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

The balanced equation is

${\text{3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO}}_{3}$