The simplest explanation for this is that the concentration of a solid can be thought of as a constant.
The equilibrium constant refers to the product of the concentration of the ions that are present in a saturated solution of an ionic compound. In solution equilibrium equations we are looking at a ratio of the ionic parts of a compound and the whole part. They may or may not have been involved in a reaction. Reaction equilibrium equations are similar in form, but different in components (reactants and products), so don’t confuse them.
The simplest explanation for this is that the concentration of a solid can be thought of as a constant. Rather than have an expression with two constants in it (the equilibrium constant and the concentration of the solid), the constants are merged to give a single value - the solubility product. See: http://www.chemguide.co.uk/physical/ksp/introduction.html
In writing out a chemical reaction equation the state of the compound (solid, gas, liquid) may be included. Ionic equations may separate the ions in solution for ease of manipulation or study, but all of the compounds or elements are always present in the equation. See also:
In a solubility equilibrium equation we are only looking at the amounts of ionized or solubilized compound in the solution compared to the solid or precipitate remaining and not in solution.
When we say that a compound is “soluble” we mean that a significant proportion of the original compound has gone into the solution.