https://file.aiquickdraw.com/mathbot/file/1727285469065jvk1wrv1.jpg

Let's analyze the problem in the image and solve it step by step:

The task asks to:
1. Find the linear equation from the provided data.
2. Predict the TOEFL score for the year 2029 if no improvements are made.

The data given is as follows:

| Year | TOEFL Score |
|------|-------------|
| 2009 | 382         |
| 2014 | 522         |
| 2019 | 552         |

### Step 1: Find the linear equation.
We need to establish a linear relationship between the year and the TOEFL score, \( y = mx + c \), where:
- \( y \) is the TOEFL score,
- \( x \) is the year,
- \( m \) is the slope (rate of change of the TOEFL score over time),
- \( c \) is the y-intercept.

To find the slope \( m \), we can use the formula for the slope between two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate \( m \) using the points (2009, 382) and (2019, 552):

\[
m = \frac{552 - 382}{2019 - 2009} = \frac{170}{10} = 17
\]

Now that we have \( m = 17 \), we can use one of the points to find the y-intercept \( c \). Let's use the point (2009, 382):

\[
y = mx + c \implies 382 = 17(2009) + c
\]
\[
382 = 34153 + c \implies c = 382 - 34153 = -33771
\]

Thus, the linear equation is:
\[
y = 17x - 33771
\]

### Step 2: Predict the TOEFL score in 2029.
Now, substitute \( x = 2029 \) into the linear equation:

\[
y = 17(2029) - 33771 = 34493 - 33771 = 722
\]

So, if no improvements are made, the predicted TOEFL score in 2029 is **722**.

---

Would you like further clarification on any step?

Here are 5 related questions for deeper understanding:
1. How do you interpret the slope in a linear equation?
2. How can we verify the accuracy of this linear model?
3. What does the y-intercept represent in this context?
4. How would the equation change if the trend were non-linear?
5. Can this model be applied to other types of data, like population growth?

**Tip:** Always check if the data follows a consistent trend before assuming linearity.

Determine the linear equation from the given TOEFL scores over the years and predict the score for 2029.

Explore how to use linear equations to predict TOEFL scores from historical data, covering concepts like slopes and y-intercepts with practical examples. Suitable for college-level students.

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