Predicting precipitates

Consider mixing solutions of two soluble ionic solids AX and BY.
AX exists as A+ and X ions in solution. 
BY exists as B+ and Y ions in solution.
The mixed solution contains two cations (A+ and B+) and two anions (X and Y). 

What are the possible precipitates?
Precipitates are insoluble ionic compounds and have no overall charge.
The two combinations that can form a precipitate are AY and BX.
These each have the cation of one of the solids in solution and the anion of the other. 

Precipitation occurs if one of the possible products has the cation and the anion of an insoluble solid.

What happens to the other pair of ions?
These remain in solution.
The precipitate can be separated from the solution by filtration. 
The soluble substance can be recovered by evaporation of the filtrate.