The following table represents the total cost, in dollars (y) to join a gym for x number of months. the cost includes a one-time joining fee of 10$. does the data in the table represent a proportional relationship or a non-proportional relationship? how do you know? x= 1 2 3 4 5 y= 25 40 55 70 85

let y' is the monthly cost to join the gym
∵ The cost includes a one-time joining fee 10$
∵ y' = y -10
                       for x = 1 ⇒   y' = 25 - 10 = 15  ⇒⇒⇒ (y'/x) = 15
                       for x = 2 ⇒   y' = 40 - 10 = 30  ⇒⇒⇒ (y'/x) = 15
                       for x = 3 ⇒   y' = 55 - 10 = 45  ⇒⇒⇒ (y'/x) = 15
                       for x = 4 ⇒   y' = 70 - 10 = 60  ⇒⇒⇒ (y'/x) = 15
                       for x = 5 ⇒   y' = 85 - 10 = 75  ⇒⇒⇒ (y'/x) = 15

∵ y'/x  = constant = 15  ⇒⇒⇒ y' = 15x
 
∴ y = y' + 10
   y = 15x +10

The last equation in the form ⇒⇒⇒ y = m x + c
Which represents a non-proportional relationship
So,
       The data in the table represent a non-proportional relationship.

Another solution:

A proportional relationship has the form y = m x
Or (y/x) = m = constant
So,
       By dividing the values of y by the values of x we will not get a constant value

  ∴   The data in the table represent a non-proportional relationship.

step by step solution :

step   1   :

1.1     4 = 22

(4)3 = (22)3 = 26

equation at the end of step   1   :

  ((23) • 26)2

step   2   :

equation at the end of step   2   :

  (23 • 26)2

step   3   :

final result :

  218

ya