https://file.aiquickdraw.com/mathbot/file/1730148792107fw49mcwc.jpg

To analyze this problem, let's interpret the table and find the relevant information about the linear relationship.

The table provides:
- **Number of classes taken (x)** and
- **Total cost (y)**

Given points from the table:
1. $ (4, 180) $
2. $ (8, 320) $
3. $ (10, 390) $
4. $ (16, 600) $

### Step 1: Determine the Slope (Rate per Class)
Since this is a linear relationship, we can calculate the slope $ m $, which represents the cost per class, using two points. Let's use points $ (4, 180) $ and $ (8, 320) $.

The formula for slope $ m $ is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using $ (x_1, y_1) = (4, 180) $ and $ (x_2, y_2) = (8, 320) $:
\[
m = \frac{320 - 180}{8 - 4} = \frac{140}{4} = 35
\]
So, the cost per class is **$35**.

### Step 2: Determine the Y-intercept (Membership Fee)
Now that we know the slope, we can use the equation of a line, $ y = mx + b $, to find the y-intercept $ b $, where $ b $ represents the membership fee.

Using the point $ (4, 180) $ and substituting $ m = 35 $:
\[
180 = 35(4) + b
\]
\[
180 = 140 + b
\]
\[
b = 180 - 140 = 40
\]
So, the membership fee is **$40**.

### Conclusion
The equation for the total cost $ y $ as a function of the number of classes $ x $ is:
\[
y = 35x + 40
\]
The correct answer to the question is that the **y-intercept is $40**, which represents the membership fee.

Would you like more details on any of these steps or have additional questions?

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Here are 5 related questions:
1. What would be the total cost for taking 20 classes?
2. How would the equation change if the membership fee increased to $50?
3. If a person only pays the membership fee, what would their total cost be?
4. How many classes can one take for a total cost of $500?
5. What is the slope-intercept form of a line, and why is it useful in this context?

**Tip:** Remember that in a linear cost relationship, the y-intercept often represents fixed costs, like membership fees or base charges.

A fitness center charges a membership fee and also charges a fee for each strength and cardio class. This table shows the linear relationship between the number of month-long classes taken and the total cost, which includes the membership fee. If the data in the table is graphed, which choice is true?

Explore the linear relationship between the number of classes taken and the total cost at a fitness center, including membership fees and class costs. Learn how to determine the y-intercept representing the membership fee and calculate the per-class cost. Suitable for students in Grades 8-10.

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