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, which is the equation of the line? (1 point) \\(\circ\\) \\(y = 200x + 9900\\) \\(\circ\\) \\(y = 9900x - 200\\) \\(\circ\\) \\(y = -200x + 9900\\) \\(\circ\\) \\(y = -200x + 9300\\) graphing calculator

Explanation:

Step1: Define variables and points

Let \( x \) be the number of years after 2010, and \( y \) be the population. We have two points: \((3, 9300)\) and \((9, 8100)\).

Step2: Calculate the slope \( m \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting the points:
\( m=\frac{8100 - 9300}{9 - 3}=\frac{- 1200}{6}=-200 \)

Step3: Use point - slope form to find the equation

Using the point \((3, 9300)\) and the slope \( m = - 200\), the point - slope form is \( y - y_1=m(x - x_1) \).
\( y-9300=-200(x - 3) \)
Expand the right - hand side: \( y-9300=-200x + 600 \)
Add 9300 to both sides: \( y=-200x+600 + 9300=-200x + 9900 \)

We can also check by substituting \( x = 3\) into \( y=-200x + 9900\): \( y=-200\times3 + 9900=-600 + 9900 = 9300\), which matches. Substituting \( x = 9\): \( y=-200\times9+9900=-1800 + 9900 = 8100\), which also matches.

Answer:

\( y=-200x + 9900 \) (the third option: \( y = - 200x+9900 \))