Question

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: Calculate the cost per visit

Let \(x\) be the cost per visit.
The number of additional visits from \(5\) to \(11\) is \(11 - 5=6\).
The amount of money spent in these \(6\) visits is \(123.75 - 74.25 = 49.5\).
So, the cost per visit \(x=\frac{123.75 - 74.25}{11 - 5}=\frac{49.5}{6}=8.25\)

Step2: Calculate the one - time sauna fee

Let \(y\) be the one - time sauna fee.
Using the information after \(5\) visits: \(180-5x=y + 123.75\)
Substitute \(x = 8.25\) into the equation: \(180-5\times8.25=y + 123.75\)
\(180 - 41.25=y + 123.75\)
\(138.75=y + 123.75\)
\(y=138.75 - 123.75=15\)

Step3: Write the linear equation

Let \(V\) be the number of visits and \(C\) be the remaining cost on the card.
The total cost of the card is \(180\). The cost of \(V\) visits is \(8.25V\), and the one - time sauna fee is \(15\).
The equation is \(C=180-(8.25V + 15)\), which simplifies to \(C = 165-8.25V\)

Answer:

\(C = 165-8.25V\) (where \(C\) is the remaining amount on the card and \(V\) is the number of visits)