Nakeisha is signing up for a gym membership with a one-time fee to join and then a monthly fee to remain a member. Let CC represent the total cost of the gym membership over tt months. The table below has select values showing the linear relationship between tt and C, .C. Determine the one-time joining fee.

To determine the one-time joining fee, we need to examine the linear relationship between the total cost \( C \) and the number of months \( t \). The general form of the linear equation is:

\[
C = mt + b
\]

where:
- \( C \) is the total cost after \( t \) months,
- \( m \) is the monthly fee,
- \( t \) is the number of months, and
- \( b \) is the one-time joining fee.

To find \( b \), we can use two points from the table to first determine \( m \), the slope, and then solve for \( b \).

Given points:
- \( (t_1, C_1) \)
- \( (t_2, C_2) \)

The slope \( m \) can be found using:

\[
m = \frac{C_2 - C_1}{t_2 - t_1}
\]

Once we have \( m \), we substitute it into the linear equation along with one of the points to solve for \( b \).

If you provide the specific values from the table, I can calculate the one-time joining fee for you.

Do you want more details or have any questions? 

Here are five related questions:
1. How does the one-time fee affect the overall cost over time?
2. If the monthly fee increases, how will that change the linear equation?
3. What is the interpretation of the slope in this context?
4. How would the total cost change if the membership duration doubles?
5. How can we predict the total cost after any number of months using this linear equation?

**Tip:** Understanding how to determine the slope of a line is key in interpreting linear relationships in real-life contexts.

Learn how to determine the one-time joining fee for a gym membership based on a linear relationship between total cost and months of membership. Understand the calculation process using two points from the provided table. Suitable for students in Grades 7-9.

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