# 13.3Linear Equations

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

$y = a + bx y=a+bx$

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.

$y=3+2xy=3+2x$
$y=–0.01+1.2xy=–0.01+1.2x$

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 + 2x.

Figure 13.3

## Try It 13.2

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

Figure 13.4

## Example 13.3

Aaron's Word Processing Service (AWPS) 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.

### Problem

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

## 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".

Figure 13.5 Three possible graphs of y = a + bx. (a) If b > 0, the line slopes upward to the right. (b) If b = 0, the line is horizontal. (c) If b < 0, the line slopes downward to the right.

## 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 + 15x.

### Problem

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