Question

1. use patterns and structure jeffrey purchased a card for $180 that gives him 20 visits to a new gym and includes a one - time fee for unlimited use of the sauna. after 5 visits, jeff has $123.75 left on the card, and after 11 visits, he has $74.25 left on the card. write an equation that jeffrey can use to determine the cost of each visit and the fee for the sauna use.

Explanation:

Step1: Find the cost per visit

Let the cost per visit be $x$. The change in the number of visits from 5 to 11 is $11 - 5=6$ visits, and the change in the card - balance is $123.75 - 74.25 = 49.5$ dollars. Using the formula for the slope (cost per visit), we have $x=\frac{123.75 - 74.25}{11 - 5}$.
$x=\frac{49.5}{6}=8.25$ dollars per visit.

Step2: Find the one - time sauna fee

Let the one - time sauna fee be $y$. We know that the initial cost of the card is $180$ dollars for 20 visits and the sauna fee. Using the information after 5 visits, the amount spent in 5 visits is $5x$. The initial amount ($180$) minus the amount spent in 5 visits and the remaining balance ($123.75$) gives the sauna fee. So, $y = 180-5\times8.25 - 123.75$.
First, calculate $5\times8.25 = 41.25$. Then $180-41.25 - 123.75=15$ dollars.
Let $C$ be the remaining balance on the card and $n$ be the number of visits. The equation is $C = 180-(8.25n + 15)=165 - 8.25n$.

Answer:

$C = 165-8.25n$, where $C$ is the balance on the card and $n$ is the number of visits. The cost per visit is $8.25$ dollars and the one - time sauna fee is $15$ dollars.