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

where *a* and *b* are constant numbers.

The variable ** x is the independent variable, and y is the dependent variable.** Another way to think about this equation is a statement of cause and effect. The X variable is the cause and the Y variable is the hypothesized effect. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

## Example 13.1

The following examples are linear equations.

## Try It 13.1

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 13.2

Graph the equation *y* = –1 + 2*x*.

## Try It 13.2

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

## Example 13.3

A local small business completes federal tax returns for customers. 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.

### Problem

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

### Solution

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

Let *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 tax return processing only. The total cost is: *y* = 31.50 + 32*x*

## Try It 13.3

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.

## Slope and *Y*-Intercept of a Linear Equation

For the linear equation *y* = *a* + *bx*, *b* = slope and *a* = *y*-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the *y*-intercept is the *y* coordinate of the point (0, *a*) where the line crosses the *y*-axis. From calculus the slope is the first derivative of the function. For a linear function the slope is *dy* / *dx* = *b* where we can read the mathematical expression as "the change in *y* (*dy*) that results from a change in *x* (*dx*) = *b* * *dx*".

## Example 13.4

Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is *y* = 25 + 15*x*.

### Problem

What are the independent and dependent variables? What is the *y*-intercept and what is the slope? Interpret them using complete sentences.

### Solution

The independent variable (*x*) is the number of hours Svetlana tutors each session. The dependent variable (*y*) is the amount, in dollars, Svetlana earns for each session.

The *y*-intercept is 25 (*a* = 25). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when *x* = 0). The slope is 15 (*b* = 15). For each session, Svetlana earns $15 for each hour she tutors.

## Try It 13.4

Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is *y* = 25 + 20*x*.

What are the independent and dependent variables? What is the *y*-intercept and what is the slope? Interpret them using complete sentences.