Question

The population of a town was taken in 2010, After 3 years, the population of the town is 9,300. After 9 years, the population of the town is 8,100

Assuming the change in population is constant each year, what is the equation of the line

Answer 1

To find the equation of the line that represents the population of the town over time, we can start by identifying two key points based on the information given:

1. Let \( t = 0 \) represent the year 2010.
- Population at \( t = 3 \) (2013) is \( 9,300 \): Point \( (3, 9300) \)

2. Population at \( t = 9 \) (2019) is \( 8,100 \): Point \( (9, 8100) \)

Next, we can find the slope \( m \) of the line using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values from our points:

- \( (x_1, y_1) = (3, 9300) \)
- \( (x_2, y_2) = (9, 8100) \)

Now plugging in:

\[
m = \frac{8100 - 9300}{9 - 3} = \frac{-1200}{6} = -200
\]

The slope \( m = -200 \) indicates that the population decreases by 200 people each year.

Next, we can use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

We can use one of our points, say \( (3, 9300) \):

\[
y - 9300 = -200(x - 3)
\]

Now, we can simplify this to put it in slope-intercept form \( y = mx + b \):

\[
y - 9300 = -200x + 600
\]

Adding \( 9300 \) to both sides:

\[
y = -200x + 9900
\]

Thus, the equation representing the population \( y \) in terms of the years \( t \) (where \( t \) is the number of years after 2010) is:

\[
y = -200t + 9900
\]

This is the equation of the line representing the population of the town over time.