Chapter 10: Linear Regression and Correlation
10.1 Linear Equations
Learning Objectives
By the end of this section, the student should be able to:
- Interpret the independent and dependent variable
- Calculate and interpret the slope and y intercept of a linear equation
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. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
Example
The following examples are linear equations.
Try It
Is the following an example of a linear equation?
y = –0.125 – 3.5x
Solution
yes
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.
Try It
Graph the equation y = –1 + 2x.
Try It
Is the following an example of a linear equation? Why or why not?
Solution
No, the graph is not a straight line; therefore, it is not a linear equation.
Try It
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.
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 word processing only. The total cost is: y = 31.50 + 32x
Try It
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.
Solution
y = 50 + 20x
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.
Example
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.
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
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 + 20x.
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 Ethan works each visit. The dependent variable (y) is the amount, in dollars, Ethan earns for each visit.
The y-intercept is 25 (a = 25). At the start of a visit, Ethan charges a one-time fee of 💲25 (this is when x = 0). The slope is 20 (b = 20). For each visit, Ethan earns 💲20 for each hour he works.
References
Data from the Centers for Disease Control and Prevention.
Data from the National Center for HIV, STD, and TB Prevention.
