Question

At one gym, there is a $12 start-up fee, and after that each month at the gym costs $20. At another gym, it costs $4 to startup, but after that, each month at the gym costs $22. After how many months will the cost of both gyms be thge same?

Answer 1

In 4 months the cost of both gyms will be the same.

Explanation:

At first we need to model the function to calculate the cost of the 2 gyms.

Slope-intercept equation of linear function

`f(x)=mx+b`

where

`m\rightarrow` slope of line

`b\rightarrow` y-intercept

Let linear function to calculate total cost of gym be:

`c(n)=mn+b`

where

`c\rightarrow` total cost of gym

`m\rightarrow` cost per month (slope)

`n\rightarrow` number of months

`b\rightarrow` start-up fee (y-intercept)

For Gym 1

`m=20` ,
`b=12`

`c(n)=20n+12`

For Gym 2

`m=22` ,
`b=4`

`c(n)=22n+4`

In order to find the number of months the cost of both gyms will be the same, we need to equate both functions and solve for number of months
`(n)`

`20n+12=22n+4`

`\\\textrm{Subtracting 22n from both sides}\\20n-22n+12=22n-22n+4\\-2n+12=4\\\textrm{Subtracting 12 from both sides}\\-2n+12-12=4-12\\-2n=-8\\\textrm{Dividing both sides by -2}\\(-2)/(-2)n=(-8)/(-2)\\n=4\textrm{ months}`

So,

In 4 months the cost of both gyms will be the same.