https://file.aiquickdraw.com/mathbot/file/1728702740400axwmjaaq.jpg

Let's break down the problem shown in the image step by step, and I'll guide you through each part:

### Given Information:
- In 2010, the annual fees for attending a public university were $7000.
- The cost increases annually, as shown in the graph.
- The graph suggests a linear increase with time, where $x$ represents the number of years after 2010.
- From the graph:
  - At $x = 0$ (the year 2010), the cost is $7000.
  - At $x = 1$ (the year 2011), the cost is $7700.

Now let's solve each part.

### Part a: Write an equation to describe the situation.
The situation appears to show a linear relationship between the cost of attending university and the number of years after 2010. The equation of a line is given by:

\[
y = mx + b
\]

Where:
- $y$ is the cost of attending university.
- $x$ is the number of years after 2010.
- $m$ is the rate of increase (slope).
- $b$ is the initial cost (the cost in 2010).

From the graph:
- The point $(0, 7000)$ gives us $b = 7000$.
- The point $(1, 7700)$ gives us the second point, which helps calculate the slope $m$.

To find the slope $m$:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7700 - 7000}{1 - 0} = \frac{700}{1} = 700
\]

Thus, the slope $m = 700$. The equation is:

\[
y = 700x + 7000
\]

### Part b: Use this model to predict the cost of attending a public university in the first year you would be eligible to enroll.
Assuming you would enroll in 2024 (which is 14 years after 2010), substitute $x = 14$ into the equation:

\[
y = 700(14) + 7000 = 9800 + 7000 = 16800
\]

The cost of attending in 2024 would be $16,800.

### Part c: What was the cost in 2000, assuming the rate of increase was the same during the time period from 2000 to 2010?
To find the cost in 2000, we need to calculate it for $x = -10$ (since 2000 is 10 years before 2010):

\[
y = 700(-10) + 7000 = -7000 + 7000 = 0
\]

The model predicts that the cost in 2000 was $0, which suggests that this linear model only holds after 2010 and might not be realistic before that.

### Part d: Are you confident with your prediction in part (c)? Explain.
No, the prediction that the cost was $0 in 2000 seems unrealistic. This indicates that the model used (a simple linear increase) does not accurately reflect the cost before 2010. There likely was a different growth pattern or a different starting value before 2010, which this model doesn't account for.

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Do you have any further questions or would you like additional details on any part of the solution?

### Related Questions:
1. How can we modify the model to account for earlier years, like 2000?
2. What other types of models (e.g., exponential) could describe university costs more accurately?
3. How does the rate of increase ($700/year) compare with inflation rates over this period?
4. If the rate of increase changed after 2024, how would that impact future costs?
5. What would the model predict for the cost in 2030?

### Tip:
Always examine whether a linear model is appropriate by considering if real-world factors, like inflation or policy changes, could make the model invalid for earlier or later years.

Suppose the annual fees for attending a public university were $7000 in 2010 and the annual cost increase is shown in the graph. Use the given data to write an equation describing the situation, predict future costs, and analyze past trends.

Learn how to write a linear equation based on university tuition costs increasing over time. This problem involves calculating the slope from a graph and using the equation to predict future costs. Suitable for high school students in Grades 9-12.

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