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The problem presented involves setting up a difference equation to model the population shifts between a city and its surrounding suburbs. Let's break down the key details from the image:

### Problem Summary:
- **6% of the city's population** moves to the surrounding suburbs each year.
- **4% of the suburban population** moves to the city each year.
- Initial population in 2020:
  - **City:** 9,900,000 residents
  - **Suburbs:** 700,000 residents
- We are asked to:
  - Set up a **difference equation** to describe the situation.
  - Estimate the populations in **2022**, two years later.

### Step 1: Set up the difference equation

Let:
- \( x_0 \) represent the population vector in 2020:
  \[
  x_0 = \begin{pmatrix} 9,900,000 \\ 700,000 \end{pmatrix}
  \]
- \( x_1 \) represent the population vector in 2021 (after one year):
  \[
  x_1 = M \cdot x_0
  \]
  where \( M \) is the matrix representing the population transitions.

Given the movement:
- **6% of the city's population moves to the suburbs**: This means **94%** of the city's population remains in the city.
- **4% of the suburban population moves to the city**: This means **96%** of the suburban population remains in the suburbs.

Thus, the transition matrix \( M \) is:
\[
M = \begin{pmatrix}
0.94 & 0.04 \\
0.06 & 0.96
\end{pmatrix}
\]

### Step 2: Apply the difference equation

Now, the population in 2021 (after one year) is:
\[
x_1 = M \cdot x_0 = \begin{pmatrix} 0.94 & 0.04 \\ 0.06 & 0.96 \end{pmatrix} \cdot \begin{pmatrix} 9,900,000 \\ 700,000 \end{pmatrix}
\]

To get the population in 2022 (after two years), we apply the matrix again:
\[
x_2 = M \cdot x_1 = M^2 \cdot x_0
\]

Let me calculate the populations for 2021 and 2022.The estimated populations are as follows:

- In **2021**:
  - City: 9,334,000 residents
  - Suburbs: 1,266,000 residents

- In **2022**:
  - City: 8,824,600 residents
  - Suburbs: 1,775,400 residents

This shows the gradual shift of population from the city to the suburbs over time.

Would you like more details on the calculation, or do you have any questions? Here are five related questions to deepen your understanding:

1. How would the population change after 5 years using the same model?
2. What effect would increasing the suburban migration rate to 8% have on the populations in 2022?
3. Could we model the situation using differential equations instead of difference equations?
4. What assumptions might be unrealistic in this model, and how could we refine it?
5. How would changes in migration rates between the city and suburbs affect long-term population trends?

**Tip**: When setting up difference equations, always check if the model's assumptions (like constant migration rates) are realistic for long-term predictions.

In a certain region, about 6% of a city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2020, there were 9,900,000 residents in the city and 700,000 residents in the suburbs. Set up a difference equation that describes this situation, where x₀ is the initial population in 2020. Then estimate the populations in the city and in the suburbs two years later, in 2022.

Learn how to set up a difference equation to model population shifts between a city and its suburbs where 6% of the city’s population moves to the suburbs and 4% of the suburban population moves to the city each year. This problem covers key concepts in difference equations, matrix multiplication, and population modeling. Suitable for students in Grades 10-12.

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