3.2 Linear Growth

Learning Objectives

  • Apply and construct linear models using the form y = mx + b

Graphs and equations that model the real world are called mathematical models.

Linear Models

Suppose the price of gasoline $2.25 per gallon. To create a graphical model of putting gasoline in a car, we can start with a table. Let’s let x represent the number of gallons, and let y represent the price the customer will pay.

x = number of gallons y = price
0 0
1 $2.25
2 $4.50
3 $6.75
4 $9.00
5 $11.25
10 $22.50
20 $45.00

In order to create an equation that models the same scenario, we can observe the pattern from the table or the graph.

If we put 0 gallons of gas in the car, the price the customer pays is $0. If we put 1 gallon of gas in the car, the price is $2.25. If we put 2 gallons of gas in the car, the price is $4.50. We notice the pattern can be written as:  price = $2.25 times the number of gallons.

Using x and y, dropping the dollar sign, and using algebra notation, the pattern can be written as an equation:   y = 2.25x

This equation can model putting gasoline in a car for any price of gasoline:

price = price per gallon x number of gallons

Now, suppose in addition to the gasoline, a person also bought a bag of chips at the gas station.Suppose the price of gasoline is $1.90 per gallon and the chips are $1.25.

x = number of gallons y = price
0 $1.25
1 $3.15
2 $5.05
3 $6.95
4 $8.85
5 $10.75
10 $20.25
20 $39.25

In the scenario of buying chips in addition to gasoline, we have a different pattern. If we put 0 gallons of gas in the car, the price the customer pays is $1.25 (for the chips only). If we put 1 gallon of gas in the car, the price is $3.15 (1 gallon of gas plus the bag of chips). If we put 2 gallons of gas in the car, the price is $5.05 (2 gallons of gas plus the bag of chips).

We notice the pattern can be written as:  price = $1.90 times the number of gallons plus $1.25

Using x and y, dropping the dollar sign, and using algebra notation, the pattern can be written as an equation:   y = 1.90x + 1.25

Try It

Suppose gasoline is $2.10 per gallon. A customer also purchases a soda pop for $1.25. Write the equation that models this scenario, where x = the number of gallons of gasoline and y = the price the customer will pay.

Let’s return to the model: y = 1.90x + 1.25

There are three terms in the model:

  • y is the price the customer will pay
  • 1.90x tells us that for every gallon of gasoline put in the car, the customer will pay $1.90.
  • 1.25 is the constant part and does not vary.

In some models, the constant term is called the initial amount or starting amount.

Consider the scenario of a student saving money to pay for tuition. Suppose the student was given $100 as birthday money. Then, the student saved $25 per week. Let y = total amount saved, and x = the number weeks since beginning to save, the model would be: y = 25x + 100.

The general model for saving money would be:

y = amount saved per week times the number of weeks plus the initial amount.

Try It

Suppose your friend was saving money to buy a car. She started with $400 and is now saving $75 per week from her part-time job. Let y = total amount saved, and x = the number of weeks since beginning to save. Write the equation that models this scenario.

Key Takeaways

To model scenarios involving linear growth, use y = mx + b.

  • In algebra, m is called the slope.
    • In our models, m was the price per gallon, or amount saved per week.
  • In algebra, b is called the y-intercept.
    • In our models, b was the price of the snack purchased at the gas station, or the starting amount in a savings plan.
  • The term mx models the part of the scenario that changes (the variable part).
  • The term b models the constant or fixed part of the scenario, which does not change.

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