# Linear Models

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

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## 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\[\begin{align} 100000 =& 10000t + 50000 \\ 50000 =& 10000t \\ 5 =& t. \end{align}\]

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?