# Linear Models

#### Contents

## Summary

Varying quantities are often modeled by **linear models**. In a linear model, the quantity increases or decreases linearly (or arithmetically in contrast to geometrically) with a variable like time. For example, we can model the position of a car moving at a constant velocity as a linear model.

As its name implies, a linear model is represented by a line on a plane, like in the figure above. A linear model is usually described by two parameters: the slope and the \(y\)-intercept. Given the slope \(a\) and the \(y\)-intercept \(b,\) the linear model can be written as a linear function \(y = ax + b.\) For the example in the figure above, \(a = \frac{4}{3}\) and \(b = 1.\)

Some authors may define a linear function as a function that satisfies

\[f(c_1 x_1 + c_2 x_2 ) = c_1 f(x_1 ) + c_2 f(x_2)\]

for all \(c_1, c_2, x_1, x_2,\) and the function \(y = ax + b,\) \(b \ne 0,\) may be called affine, but we do not follow this definition herein.

## Example Problems

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

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

A school has 100 first grade students and a class consists of 20 students. The school plans to open a new first grade class every year. Then, how many first graders will the school have after six years?

After \(t\) years, the number \(y\) of students in first grade is \(y = 20t + 100.\) Plugging in \(t = 6\) gives

\[\begin{align} y =& 20 \times 6 + 100\\ =& 120 + 100 \\ =& 220. \end{align}\]

Therefore, the school will have 220 first graders. \( _\square \)

## Suppose your bank account balance \(y\) (in dollars) after \(t\) years is given by \(y = 10000t + 50000.\) How many years do you need to 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. \( _\square \)