On ITV's Homework Hotline, it's time for "Rudolf's Rules." If you guess what rule Rudolf is using you get a prize. Someone puts golf balls into a miniature-golf display called Rudolf, then Rudolf sends back a certain number of golf balls. Today Rudolf gets 8 balls and sends back 6. Next Rudolf gets 15 and sends back 27. When he gets 25, 57 are returned. If n represents the number of balls initially given to Rudolf, write an expression to represent how many balls he sent back.

y = 3x - 18 (where y = balls sent back by Rudolf; x = balls given to Rudolf)

Step-by-step explanation:

x (Balls given) y (Balls sent back)

8 6 Δy/Δx = (27-6) / (15-8) = 3

15 27

25 57 Δy/Δx = (57-27) / (25-15) = 3

Observing the number sequence shows constant gradient and as such we can use equation of a straight line to relate the two variables.

Using the equation of a straight line we have:

(y-y1) / (x-x1) = (y2-y1) / (x2-x1)

y-6/x-8 = (27-6) / (15-8)

y-6/x-8 = 3

y-6 = 3 (x-8)

y-6 = 3x - 24

y = 3x-18