 # 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.

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.

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

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