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How Do You Use the Graph of a Linear Equation to Solve a Word Problem?

The rabbit population in South Florida peaked in 2005 at 4 million rabbits. Since then, the population has decreased by 250,000 rabbits per year. Let x represent the number of years since 2005 and y represent the total number of rabbits. The equation y = 4,000,000 – 250,000x describes the rabbit population at any time following 2005. Use a graph to figure out how long (in years) it will take for the rabbit population to go extinct, assuming no rabbits are born or brought into South Florida after 2005.

Summary

  1. 'x' represents the number of years since 2005
  2. 'y' represents the total rabbit population
  3. We're using 'M' as shorthand for 'million'
  4. The numbers along the x-axis represent the number of years since 2005
  5. The numbers along the y-axis represent the total rabbit population
  6. The point where y=0 tells us how long it will take for the rabbits to go extinct
  7. x is 16 when y is 0, so it will take 16 years for the rabbits to go extinct

Notes

    1. 'x' is a variable that represents the number of years since 2005
    2. 'y' is a variable that represents the total rabbit population
    1. We can create a table to find some ordered pairs so we can make a graph
    2. Then we'll be able to use the graph to try to solve our problem!
    1. 'x' is a variable that represents the number of years since 2005
    2. 'y' is a variable that represents the total rabbit population
    3. We were given the equation y=4,000,000-250,000x to represent the total rabbit population x years after 2005
    1. 'x' is the variable representing the number of years it's been since 2005
    1. 'x' and 'y' are variables
    2. 'x' represents the number of years it's been since 2005
    3. 'y' represents the total rabbit population
    1. 'y' represents the total rabbit population
    2. We can replace 'x' in the equation with each of the values we chose in order to find how many rabbits there were each year
    1. 'y' represents the number of rabbits
    2. The equation we started with was 'y=' something, so when we plug in we get back our values for y
    1. Ordered pairs give us instructions for how to plot points in the coordinate plane
    2. They are written in the form (x,y)
    3. If we turn the x- and y-values we just found into ordered pairs, we can make a graph for the equation!
    1. The numbers along the x-axis represent the number of years since 2005
    2. The x's range from 0 to 12, so we can make the x-axis go up to 18
    3. The numbers along the y-axis represent the total rabbit population
    4. The y's range from 4 million to 1 million, so we can make the y-axis go up to 4 million
    1. The x's range from 0 to 12, so we can make the x-axis go up to 18
    1. The y's range from 4 million to 1 million, so we can make the y-axis go up to 4 million
    1. This ordered pair means that there were 4,000,000 rabbits in 2005
    1. (4, 3 million) means that there were 3 million rabbits in 2009, 4 years after 2005
    2. (8, 2 million) means that there will be 2 million rabbits in 2013
    3. (10, 1.5 million) means that there will be 1.5 million rabbits in 2015, 10 years after 2005
    4. (12, 1 million) means that there will be 1 million rabbits in 2017, 12 years after 2005
    1. We've made a graph, but now what?
    1. 'y' represents the rabbit population
    2. So we need to find the point on the graph where y=0
    1. Remember, we're looking for the number of years it'll take for the rabbits to become extinct
    1. Since we have a linear equation, we can just draw a line through the dots
    1. This is the point where y, the total rabbit population, equals 0
    2. That means after 16 years there will be no rabbits left
    1. The rabbit population is extinct when y=0
    2. It will be 2021, 16 years after 2005, when the rabbits go extinct