# 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:

*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.

### Try It

Is the following an example of a linear equation?

*y* = –0.125 – 3.5*x*

## 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 + 2*x*.

### 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

*y*= 25 + 15

*x*.

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 + 20*x*.

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 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.

# Concept Review

The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form ** y = mx + b**, where

**and**

*m***are constants,**

*b***is the independent variable,**

*x***is the dependent variable. In a statistical context, a linear equation is written in the form**

*y***, where**

*y = a + bx***and**

*a***are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation**

*b**y = a + bx*, the constant

*b*that multiplies the

**variable (**

*x**b*is called a coefficient) is called as the

**slope**. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation

*y = a + bx*, the constant a is called as the

*y*-intercept. Graphically, the

*y*-intercept is the

*y*coordinate of the point where the graph of the line crosses the

*y*axis. At this point

*x*= 0.

The **slope of a line** is a value that describes the rate of change between the independent and dependent variables. The **slope** tells us how the dependent variable (*y*) changes for every one unit increase in the independent (*x*) variable, on average. The ** y-intercept** is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.

# Formula Review

*y* = *a* + *bx* where *a* is the *y*-intercept and *b* is the slope. The variable *x* is the independent variable and*y* is the dependent variable.