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

[y = a + b ext{x} onumber ]

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 (PageIndex{1})

The following examples are linear equations.

[y = 3 + 2 ext{x} onumber ]

[y = -0.01 + 1.2 ext{x} onumber ]

Exercise (PageIndex{1})

Is the following an example of a linear equation?

[y = -0.125 - 3.5 ext{x} onumber ]

**Answer**yes

The graph of a linear equation of the form (y = a + b ext{x}) is a **straight line**. Any line that is not vertical can be described by this equation.

Example (PageIndex{2})

Graph the equation (y = -1 + 2 ext{x}).

**Figure (PageIndex{1}).**

Exercise (PageIndex{2})

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

**Figure (PageIndex{2}).**

**Answer**No, the graph is not a straight line; therefore, it is not a linear equation.

Example (PageIndex{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.

**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 + 32 ext{x})

Exercise (PageIndex{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.

**Answer**(y = 50 + 20 ext{x})

## Slope and Y-Intercept of a Linear Equation

For the linear equation (y = a + b ext{x}), (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.

* Figure (PageIndex{3}).* Three possible graphs of (y = a + b ext{x}) (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 (PageIndex{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 + 15 ext{x}).

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

**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.

Exercise (PageIndex{4})

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 ext{x}).

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

**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.

## Summary

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 (m) and (b) are constants, (x) is the independent variable, (y) is the dependent variable. In a statistical context, a linear equation is written in the form (y = a + bx), where (a) and (b) are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation (y = a + b ext{x}), the constant b that multiplies the (x) variable ((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 + b ext{x}), 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 + b ext{x}) where *a* is the (y)-intercept and (b) is the slope. The variable (x) is the independent variable and (y) is the dependent variable.