Statistics

# 12.1Linear Equations

Statistics12.1 Linear Equations

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

$y = a + b x y=a+bx$

where 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.

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

### Try It 12.1

Is the following an example of a linear equation?

y = –0.125 – 3.5x

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 + 2x.

Figure 12.2

### Try It 12.2

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

Figure 12.3

### 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.

#### Problem

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

### Try It 12.3

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-interceptof a Linear Equation

For the linear equation y = a + bx, b = slope and a = y-inttercept. From algebra, recall that the slope is a number that describes the steepness of a line; the y-intercept is the y-coordinate of the point (0, a), where the line crosses the y-axis.

Please note that in previous courses you learned $y=mx+b y=mx+b$ was the slope-intercept form of the equation, where m represented the slope and b represented the y-intercept. In this text, the form $y=a+bx y=a+bx$ is used, where a is the y-intercept and b is the slope. The key is remembering the coefficient of x is the slope, and the constant number is the y-intercept.

Figure 12.4 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 12.4

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.

### Try It 12.4

Ethan repairs household appliances such as 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 + 20x.

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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