47 12.1 Linear Equations
Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:
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.
Try It
Is the following an example of a linear equation?
y = –0.125 – 3.5x
Show Answer
Yes
The graph of a linear equation of the form y = a + bx is a straight line.
Any straight line that is not vertical can be described by this equation.
Graph the equation y = –1 + 2x.
Try It
Is the following an example of a linear equation? Why or why not?
Show Answer
No, the graph is not a straight line; therefore, it is not a linear equation.
Example 2
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.
Show Answer:
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.
Show Answer
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 3
What are the independent and dependent variables? What is the y-intercept and what is the slope? Interpret them using complete sentences.
Show Answer
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.
Show Answer
The independent variable (x) is the number of hours Ethan wor