You start out with:

H₃PO₄ + Mg(OH)₂ –> Mg₃(PO₄)₂ + H₂O

Stay away from the H’s and O’s at first, because they appear in multiple substances on at least one side of the equation. Thus, we start with P and Mg. To balance P, multiply H₃PO₄ by 2:

2H₃PO₄ + Mg(OH)₂ –> Mg₃(PO₄)₂ + H₂O

To balance Mg, multiply Mg(OH)₂ by 3:

2H₃PO₄ + 3Mg(OH)₂ –> Mg₃(PO₄)₂ + H₂O

Now look at H’s, because they appear only once on the products side. There is 2×3 + 3×2 = 12 H’s, so multiply H₂O by 6:

2H₃PO₄ + 3Mg(OH)₂ –> Mg₃(PO₄)₂ + 6H₂O

Now the O’s are balanced, too (2×4 + 3×2 = 14 on reactants side, 4×2 +6×1 = 14 on products side).