The total cost, y, for x passes to the State Fair includes a flat fee for processing and a cost per pass. One family purchased 5 passes for a total cost of $390. Another family purchased 8 passes for $615.

Which linear equation represents the total cost in dollars, y, for x passes?

To find the linear equation that represents the total cost, $ y $, for $ x $ passes, we need to recognize that this is a linear relationship of the form:

\[
y = mx + b
\]

Where:
- $ y $ is the total cost,
- $ x $ is the number of passes,
- $ m $ is the cost per pass (slope),
- $ b $ is the flat fee for processing (y-intercept).

We are given two data points:
1. 5 passes cost $390: $(x_1, y_1) = (5, 390)$
2. 8 passes cost $615: $(x_2, y_2) = (8, 615)$

### Step 1: Find the slope $ m $
The slope $ m $ (cost per pass) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values:

\[
m = \frac{615 - 390}{8 - 5} = \frac{225}{3} = 75
\]

Thus, the cost per pass is $ 75 $ dollars.

### Step 2: Find the flat fee $ b $
Now, we can use one of the points, say $(5, 390)$, and the slope $ m = 75 $ to find the flat fee $ b $. Substituting into the equation $ y = mx + b $:

\[
390 = 75(5) + b
\]

Simplifying:

\[
390 = 375 + b
\]

\[
b = 390 - 375 = 15
\]

Thus, the flat fee is $ 15 $ dollars.

### Step 3: Write the linear equation
Now that we have both the slope $ m = 75 $ and the y-intercept $ b = 15 $, the linear equation is:

\[
y = 75x + 15
\]

This equation represents the total cost $ y $ in dollars for $ x $ passes.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this concept:
1. How would the equation change if a different flat fee were applied?
2. What would be the total cost for 10 passes?
3. How can you interpret the slope of the equation in a real-world context?
4. What happens to the total cost if the price per pass increases?
5. How can you use this equation to predict the cost for any number of passes?

**Tip:** Always check your calculations by plugging one of the given points back into the final equation to verify its accuracy.

The total cost, y, for x passes to the State Fair includes a flat fee for processing and a cost per pass. One family purchased 5 passes for a total cost of $390. Another family purchased 8 passes for $615. Which linear equation represents the total cost in dollars, y, for x passes?

Learn how to determine the linear equation representing the total cost for passes to the State Fair. This problem involves calculating the slope and y-intercept using real-world data points: 5 passes cost $390, and 8 passes cost $615. Suitable for students in Grades 7-9.

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