x + y = 2 has only one solution in positive integers which is x=1, y=1

3xy = 9 has two solutions in positive integers: x=3, y=1, and x=1, y=3

2x = 3y has an infinite number of solutions in positive integers. The 
smallest solution is x=3, y=2.  But x=1458, y=972 is also a solution. 
So is x=1427037, y=951358

7x + 3y = 13 has only one solution x=1, y=2

5x - 2y = 4 has infinitely many solutions in positive integers:x=2, y=3, and
x=4, y=8, and x=6, y=13, etc.

Edwin