The population of a city in 2003 was estimated to be 108 comma 434. By 2015​, the number had declined to 103 comma 922.
​(a)  Assuming the population​ linearly, find the linear​ model, y=mx+b, representing the population size x years since 2000.
y equals
  
negative 376 x plus 109562
​(round coefficients to the nearest whole​ number)
 
​(b) Using the linear model from part​ (a), estimate the population in 2028
  
enter your response herepeople
​(round to the nearest whole​ number)


### Part (a): Finding the linear model

Given the population data for the years 2003 and 2015:
- Population in 2003: 108,434
- Population in 2015: 103,922

We are asked to model the population as a linear function of time, where \( x \) is the number of years since 2000. So, for 2003, \( x = 3 \), and for 2015, \( x = 15 \).

The equation of a line is given by:

\[
y = mx + b
\]

We need to find the slope \( m \) and the intercept \( b \).

#### Step 1: Calculate the slope \( m \)

The slope is the rate of change of population per year and is calculated as:

\[
m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{103922 - 108434}}{{15 - 3}} = \frac{{-4512}}{{12}} = -376
\]

So, the slope \( m = -376 \).

#### Step 2: Find the intercept \( b \)

To find \( b \), we use one of the points (e.g., \( (3, 108434) \)) and the equation \( y = mx + b \).

\[
108434 = -376(3) + b
\]

\[
108434 = -1128 + b
\]

\[
b = 108434 + 1128 = 109562
\]

So, the linear model is:

\[
y = -376x + 109562
\]

### Part (b): Estimating the population in 2028

For 2028, the value of \( x \) is \( 2028 - 2000 = 28 \).

Using the linear model \( y = -376x + 109562 \), we substitute \( x = 28 \):

\[
y = -376(28) + 109562
\]

\[
y = -10528 + 109562
\]

\[
y = 99034
\]

So, the estimated population in 2028 is **99,034 people**.

---

Do you want more details or have any questions? Here are 5 related questions to explore:

1. How would the population change if the decline rate increased to -400 per year?
2. Can you find the population model if the decline were exponential instead of linear?
3. What would be the population in 2050 according to the current model?
4. How long will it take for the population to drop below 90,000?
5. How would changes in birth rate affect the linear model?

**Tip:** When estimating future values using a linear model, remember that real-world factors may cause deviations from the projected trend.

The population of a city in 2003 was estimated to be 108,434. By 2015, the number had declined to 103,922. (a) Assuming the population declined linearly, find the linear model, y = mx + b, representing the population size x years since 2000. (b) Using the linear model from part (a), estimate the population in 2028.

Explore how to model a city's population decline from 2003 to 2015 using a linear equation. Learn how to find the linear model y = -376x + 109562 and use it to estimate the population in 2028. Suitable for students in Grades 9-12 studying linear equations and population modeling.

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