Linear equations are helpful tools for understanding how populations grow in different cities. By using these equations, we can make predictions based on past information. This is really important for city planning and deciding how to use resources.

Population Growth Models

1. Creating the Equation: A simple linear equation for population growth looks like this: $P(t) = P_0 + rt$ Here’s what each part means:

   • $P(t)$ is the population at time $t$.
   • $P_0$ is the starting population.
   • $r$ is the number of new people added each year.
   • $t$ is the number of years since we started counting.

2. Example Cities and Data:

   • City A: In 2020, the population was 500,000, and it grows by 5,000 people each year. So, the equation is: $P(t) = 500000 + 5000t$
   • City B: In 2020, the population was 300,000, and it grows by 3,000 people each year. So, the equation is: $P(t) = 300000 + 3000t$

Analyzing the Data

• Predictions for City A:
  • In 2025 ($t=5$): $P(5) = 500000 + 5000(5) = 525000$
• Predictions for City B:
  • In 2025 ($t=5$): $P(5) = 300000 + 3000(5) = 315000$

Graphing the Data

When we graph these equations, we see straight lines that show how the population goes up over time. The steepness of the line shows how fast the population is growing. This makes it easy to see and compare the growth of different cities.

Conclusion

By looking at linear equations, city planners can make smart choices about what buildings and services are needed for the growing number of people.