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Statistics LibreTexts

12.2: Linear Equations

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

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

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.

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

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

Exercise \(\PageIndex{1}\)

Is the following an example of a linear equation?

\[y = -0.125 - 3.5\text{x}\]

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 \(y = -1 + 2\text{x}\).

Graph of the equation y = -1 + 2x.  This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.

Figure 12.2.1.

Exercise \(\PageIndex{2}\)

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

This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.

Figure 12.2.2.

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.

Three possible graphs of the equation y = a + bx. For the first graph, (a), b > 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b < 0 and the line slopes downward to the right.

Figure 12.2.3. 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.

References

  1. Data from the Centers for Disease Control and Prevention.
  2. Data from the National Center for HIV, STD, and TB Prevention.

Chapter 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 \(m\) 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.

 

 

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

Exercise 12.2.5

What are the dependent and independent variables?

Answer

dependent variable: fee amount; independent variable: time

Exercise 12.2.6

Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

Exercise 12.2.7

Graph the equation from Exercise.

Answer

This is a graph of the equation y = 25 + 12.50x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 25 from 0 - 100. The equation's graph is a line that crosses the y-axis at 25 and is sloped up and to the right, rising 12.50 units for every one unit of run.

Figure 12.2.4.

Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

Exercise 12.2.8

Find the equation that expresses the total fee in terms of the number of days the payment is late.

Exercise 12.2.9

Graph the equation from Exercise.

Answer

This is a graph of the equation y = 10 + 5x. The x-axis is labeled in intervals of 1 from 0 - 7; the y-axis is labeled in intervals of 10 from 0 - 50. The equation's graph is a line that crosses the y-axis at 10 and is sloped up and to the right, rising 5 units for every one unit of run.

Figure 12.2.5.

Exercise 12.2.10

Is the equation \(y = 10 + 5x – 3x^{2}\) linear? Why or why not?

Exercise 12.2.11

Which of the following equations are linear?

  1. \(y = 6x + 8\)
  2. \(y + 7 = 3x\)
  3. \(y – x = 8x^{2}\)
  4. \(4y = 8\)

Answer

\(y = 6x + 8\), \(4y = 8\), and \(y + 7 = 3x\) are all linear equations.

Exercise 12.2.12

Does the graph show a linear equation? Why or why not?

This is a graph of an equation. The x-axis is labeled in intervals of 1 from -5 to 5; the y-axis is labeled in intervals of 1 from 0 - 8. The equation's graph is a parabola, a u-shaped curve that has a minimum value at (0, 0).

Figure 12.2.6.

Table contains real data for the first two decades of AIDS reporting.

Adults and Adolescents only, United States
Year # AIDS cases diagnosed # AIDS deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Exercise 12.2.13

Use the columns "year" and "# AIDS cases diagnosed. Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?

Answer

The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable.

Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is \(y = 50 + 100x\).

Exercise 12.2.14

What are the independent and dependent variables?

Exercise 12.2.15

What is the y-intercept and what is the slope? Interpret them using complete sentences.

Answer

The \(y\)-intercept is 50 (\(a = 50\)). At the start of the cleaning, the company charges a one-time fee of $50 (this is when \(x = 0\)). The slope is 100 (\(b = 100\)). For each session, the company charges $100 for each hour they clean.

Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y = 12,000x\).

Exercise 12.2.16

What are the independent and dependent variables?

Exercise 12.2.17

How many pounds of soil does the shoreline lose in a year?

Answer

12,000 pounds of soil

Exercise 12.2.18

What is the \(y\)-intercept? Interpret its meaning.

Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is \(y = 15 – 1.5x\) where \(x\) is the number of hours passed in an eight-hour day of trading.

Exercise 12.2.19

What are the slope and y-intercept? Interpret their meaning.

Answer

The slope is -1.5 (\(b = -1.5\)). This means the stock is losing value at a rate of $1.50 per hour. The \(y\)-intercept is $15 (\(a = 15\)). This means the price of stock before the trading day was $15.

Exercise 12.2.19

If you owned this stock, would you want a positive or negative slope? Why?