# 11.1: Linear Equations

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- 21535

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

\[y = a + b\text{x}\nonumber \]

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.

\[y = 3 + 2\text{x}\nonumber \]

\[y = -0.01 + 1.2\text{x}\nonumber \]

Exercise \(\PageIndex{1}\)

Is the following an example of a linear equation?

\[y = -0.125 - 3.5\text{x}\nonumber \]

**Answer**-
yes

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

Example \(\PageIndex{2}\)

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

**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: \(y = 31.50 + 32\text{x}\)

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**-
\(y = 50 + 20\text{x}\)

## Slope and Y-Intercept of a Linear Equation

For the linear equation \(y = a + b\text{x}\), \(b =\) 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 \(y = a + b\text{x}\) (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 \(y = 25 + 15\text{x}\).

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 \(y = 25 + 20\text{x}\).

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 \(y = a + b\text{x}\), 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 \(y = a + b\text{x}\), 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

\(y = a + b\text{x}\) where *a* is the \(y\)-intercept and \(b\) is the slope. The variable \(x\) is the independent variable and \(y\) is the dependent variable.