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.

Question

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

y=−200x+9900
y is equal to negative 200 x plus 9900

y=200x+9900
y is equal to 200 x plus 9900

y=9900x−200
y is equal to 9900 x minus 200

y=−200x+9300
y is equal to negative 200 x plus 9300

Answer 1

To find the equation of the line that models the population change over time, we can start with the information provided:

• In 2013 (3 years after 2010), the population is 9,300.
• In 2019 (9 years after 2010), the population is 8,100.

We can set up two points based on this data:

• Point 1 (2013): (3, 9300)
• Point 2 (2019): (9, 8100)

Now we can calculate the slope (m) of the line using the formula:

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

Substituting the values of the two points:

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

So, the slope of the line is -200.

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

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

Using point (3, 9300) for \((x_1, y_1)\):

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

Expanding this equation:

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

Adding 9300 to both sides:

\[ y = -200x + 9900 \]

So the equation of the line is:

\[ \boxed{y = -200x + 9900} \]

This matches the first option in your list:

y = −200x + 9900