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Let's solve the problem step by step using the data provided:

### Part a) Write an equation to find the average cost, $ C $, of tuition after $ x $ years.

#### Step 1: Analyze the data.
The table provides tuition and fees over several academic years. We can treat this as a linear relationship since the increases seem consistent.

- 2012–13: $ 9006 $
- 2013–14: $ 9077 $
- 2014–15: $ 9161 $
- 2015–16: $ 9410 $

To model this with a linear equation $ C(x) = mx + b $, we need to find the slope ($ m $) and the y-intercept ($ b $).

#### Step 2: Calculate the slope $ m $.
We can use two points to calculate the slope. Let the year 2012–13 be $ x = 0 $ and each subsequent year increment $ x $ by 1.

Points:
- $ (0, 9006) $
- $ (1, 9077) $
- $ (2, 9161) $
- $ (3, 9410) $

Using the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using points $ (0, 9006) $ and $ (3, 9410) $:
\[
m = \frac{9410 - 9006}{3 - 0} = \frac{404}{3} \approx 134.67
\]
Thus, the slope $ m \approx 134.67 $.

#### Step 3: Find the y-intercept $ b $.
Using the point $ (0, 9006) $ in the equation $ C(x) = mx + b $:
\[
9006 = 134.67(0) + b \implies b = 9006
\]

#### Final equation:
The linear equation is:
\[
C(x) = 134.67x + 9006
\]
where $ x $ is the number of years after 2012.

### Part b) Predict when tuition will exceed $10,000.

We need to find $ x $ such that $ C(x) > 10000 $.

Using the equation:
\[
134.67x + 9006 > 10000
\]
Solving for $ x $:
\[
134.67x > 10000 - 9006
\]
\[
134.67x > 994
\]
\[
x > \frac{994}{134.67} \approx 7.38
\]
Since $ x $ represents years after 2012, the tuition will exceed $10,000 in approximately 7.38 years, which is sometime in the 2019–20 academic year.

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Would you like more details or have any other questions? Here are 5 related questions:

1. How can you model a nonlinear increase in tuition?
2. Can this model predict the tuition beyond 2020 accurately?
3. How would the equation change if there were larger jumps in tuition fees?
4. Can the linear equation be adjusted for inflation over time?
5. How accurate is using a linear model for tuition prediction?

**Tip**: Linear models are useful for small, consistent increases but may not hold in the long term if the rate of increase changes drastically.

The table shows the average cost of tuition and fees at a public four-year college for an in-state student. Write an equation to find the average cost, C, of tuition after x years and predict when tuition will exceed $10,000.

Solve a real-world problem of predicting tuition costs using linear equations. Analyze a table of tuition costs from 2012-2016, find the slope, and determine when tuition will exceed $10,000. Suitable for Grades 9-12 students learning about slope-intercept form.

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