Let c represent the number of chickens. Let d represent the number of ducks.

The total number of animals bought was 20. So c + d = 20.

The man received a $2 discount per chicken and a $0.50 discount per duck. The total discount he received was $22. So 2c + 0.50d = 22.

Now we have two equations with two varaibles. First we will rewrite the first equation to find the value of c.

c + d = 20

c = 20 - d

Now we can substitue the vale of c from above into the second equation and solve for d.

2c + 0.50d = 22

2(20 - d) + 0.50d = 22

40 - 2d + 0.50d = 22

40 - 1.50d = 22

-1.50d = -18

d = 12

Now solve for c in the first equation.

c + d = 20

c + 12 = 20

c = 8