The image provides a problem asking to model the population density over time as a linear equation. Here is a breakdown of how to approach it:

Problem Summary:

• In 1980 (which is considered year zero), the population density was 70 people per square mile.
• In 2000, the population density had grown to 95 people per square mile.
• You need to find a linear equation in the form $y = mx + b$, where:
  • $x$ is the time in years since 1980.
  • $y$ is the population density (in people per square mile).

Steps to Solve:

1. Identify the Two Key Points:

   • In 1980, $x = 0$ and $y = 70$.
   • In 2000, $x = 2000 - 1980 = 20$ and $y = 95$.

2. Find the Slope ($m$): The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(0, 70)$ and $(20, 95)$: $m = \frac{95 - 70}{20 - 0} = \frac{25}{20} = \frac{5}{4}$

3. Write the Equation: Since the slope is $\frac{5}{4}$ and the y-intercept (when $x = 0$) is 70, the equation is: $y = \frac{5}{4}x + 70$

This is the linear equation that models the population density.

Final Answer:

The equation is: $y = \frac{5}{4}x + 70$

Would you like any further details on the solution or have additional questions?

Related Questions:

1. How do we interpret the slope in this context?
2. How would you adjust the equation if we wanted to start from a different year, like 1990?
3. What does the y-intercept represent in this problem?
4. How would you graph this equation on a coordinate plane?
5. What would happen if the population density increased more rapidly?

Tip:

When working with linear equations, always ensure to identify your two points clearly to calculate the slope correctly before forming the equation!