# Linear Models

A **linear model** is an equation that describes a relationship between two quantities that show a constant rate of change.

#### Contents

## Summary

We represent linear relationships graphically with straight lines. A linear model is usually described by two parameters: the slope, often called the growth factor or rate of change, and the $y$-intercept, often called the initial value. Given the slope $m$ and the $y$-intercept $b,$ the linear model can be written as a linear function $y = mx + b.$

We can represent the position of a car moving at a constant velocity with a linear model.

In the figure above, the rate of change is $\frac{200 \text{ m}}{10\text{ s}},$ or $20 \text{ m/s},$ so $m=20.$ The initial value is $200 \text{ meters}$ so $b = 200.$ The linear model that represents this car's position is $y=20x + 200.$

## Writing and Using Linear Models

To write a linear model we need to know both the rate of change and the initial value. Once we have written a linear model, we can use it to solve all types of problems.

A gym has 100 members. The gym plans to increase membership by 10 members every year. Write an equation to represent the relationship between the number of members, $y,$ and the years from now, $x.$

The initial value is 100 and the rate of change is 10. Therefore, $y=10x+100.$

The table below shows the cost of an ice cream cone $y$ with $x$ toppings. Write an equation that models the relationship between $x$ and $y.$

Toppings $x$ Cost $y$ 2 $3.50 3 $3.75 5 $4.25

Adding one additional topping costs $\$3.75-\$3.50=\$0.25$ or $\frac{\$4.25-\$3.75}{2} = \$0.25.$ If an ice cream cone with two toppings costs $3.50 each topping costs $0.25, then a cone without any toppings must cost $3.00. Therefore, the rate of change is 0.25, the initial value is 3, and $y=0.25x + 3.$

The position $y$ (in kilometers) of a car at time $t$ (in hours) is given by $y = 80t + 300.$ How far does the car travel in one hour?

When $t$ increases by one hour, $y$ increases by 80 kilometers, so our answer is 80 kilometers. $_\square$

## Suppose your bank account balance $y$ (in dollars) after $t$ years is given by $y = 10000t + 50000.$ In how many years will you have 100,000 dollars?

Plugging 100,000 into $y$ gives$\begin{aligned} 100000 =& 10000t + 50000 \\ 50000 =& 10000t \\ 5 =& t. \end{aligned}$

Therefore, you need 5 years. $_\square$

Henry deposits the same amount of money into his bank account each week. After six weeks of saving money, Henry has $70 in his bank account. After ten weeks of saving money, he has $82. Which equation represents the amount of money $y$ that Henry has in his bank account after $x$ weeks?