### Introduction

Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form

*a*and

*b*are constant numbers.

The variable *x* is the *independent variable*; *y* is the *dependent variable*. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

### Example 12.1

The following examples are linear equations.

Is the following an example of a linear equation?

*y* = –0.125 – 3.5*x*

The graph of a linear equation of the form *y* = *a* + *bx* is a straight line. Any line that is not vertical can be described by this equation.

### Example 12.2

Graph the equation *y* = –1 + 2*x*.

Is the following an example of a linear equation? Why or why not?

### Example 12.3

Aaron’s Word Processing Service does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.

Find the equation that expresses the total cost in terms of the number of hours required to complete the job.

Let *x* = the number of hours it takes to get the job done.

*y*= the total cost to the customer.

The $31.50 is a fixed cost. If it takes *x* hours to complete the job, then (32)(*x*) is the cost of the word processing only. The total cost is *y* = 31.50 + 32*x*.

Emma’s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class, as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class.