Intermediate algebra with applications and visualization 3rd edition pdf

89

Source: J. Williams, The USA Today Weather Almanac.

2.1 Introduction to Equations

2.2 Linear Equations 2.3 Introduction to

Problem Solving 2.4 Formulas 2.5 Linear Inequalities

Mathematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity mea- sured in milliwatts per square meter for various latitudes and dates.

2 Linear Equations and Inequalities

Education is not the filling of a pail, but the lighting of a fire.

— WILLIAM BUTLER YEATS

If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 24999 � 2.5 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities.

Latitude Mar. 21 June 21 Sept. 21 Dec. 21

0� 325 254 325 272

10� 311 275 280 220

20� 249 292 256 143

30� 179 248 182 80

40� 99 199 127 34

50� 57 143 75 13

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

90 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

2.1 Introduction to Equations Basic Concepts ● Equations and Solutions ● The Addition Property of Equality ● The Multiplication Property of Equality

A LOOK INTO MATH N The Global Positioning System (GPS) consists of 24 satellites that travel around Earth in nearly circular orbits. GPS can be used to determine locations and velocities of cars, air- planes, and hikers with an amazing degree of accuracy. New cars often come equipped with GPS, and their drivers can determine their cars’ locations to within a few feet. To cre- ate GPS, thousands of equations were solved, and mathematics was essential in finding their solutions. In this section we discuss many of the basic concepts needed to solve equa- tions. (Source: J. Van Sickle, GPS for Land Surveyors.)

Basic Concepts N REAL-WORLD CONNECTION Suppose that during a storm it rains 2 inches before noon and

1 inch per hour thereafter until 5 P.M. Table 2.1 lists the total rainfall R after various elapsed times x, where x = 0 corresponds to noon.

NEW VOCABULARY

n Solution n Solution set n Equivalent equations

TABLE 2.1 Rainfall x Hours Past Noon

Elapsed Time: x (hours) 0 1 2 3 4 5

Total Rainfall: R (inches) 2 3 4 5 6 7

The data suggest that the total rainfall R in inches is 2 more than the elapsed time x. A formula that models, or describes, the rainfall x hours past noon is given by

R = x + 2.

For example, 3 hours past noon, or at 3 P.M.,

R = 3 + 2 = 5

inches of rain have fallen. Even though x = 4.5 does not appear in the table, we can calcu- late the amount of rainfall at 4:30 P.M. with the formula as

R = 4.5 + 2 = 6.5 inches.

The advantage that a formula has over a table of values is that a formula can be used to cal- culate the rainfall at any time x, not just at the times listed in the table.

Equations and Solutions In the rainfall example above, how can we determine when 6 inches of rain have fallen? From Table 2.1 the solution is 4, or 4 P.M. To find this solution without the table, we can solve the equation

x + 2 = 6.

An equation can be either true or false. For example, the equation 1 + 2 = 3 is true, whereas the equation 1 + 2 = 4 is false. When an equation contains a variable, the equa- tion may be true for some values of the variable and false for other values of the variable. Each value of the variable that makes the equation true is called a solution to the equation,

READING CHECK

• What is a solution to an equation?

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

912.1 INTRODUCTION TO EQUATIONS

and the set of all solutions is called the solution set. Solving an equation means finding all of its solutions. Because 4 + 2 = 6, the solution to the equation

x + 2 = 6

is 4, and the solution set is {4}. Note that braces {} are used to denote a set. Table 2.2 shows examples of equations and their solution sets. Note that some equations, such as the third one in the table, can have more than one solution.

TABLE 2.2 Equations and Solution Sets

Equation Solution Set True Equation(s)

3 - x = 1 {2} 3 - 2 = 1

10 - 4y = 6 {1} 10 - 4(1) = 6

x2 = 4 {�2, 2} (�2)2 = 4 and 22 = 4

Many times we cannot solve an equation simply by looking at it. In these situations we must use a step-by-step procedure. During each step an equation is transformed into a different but equivalent equation. Equivalent equations are equations that have the same solution set. For example, the equations

x + 2 = 5 and x = 3

are equivalent equations because the solution set for both equations is {3}.

When solving equations, it is often helpful to transform a more complicated equation into an equivalent equation that has an obvious solution, such as x = 3. The addition prop- erty of equality and the multiplication property of equality can be used to transform an equation into an equivalent equation that is easier to solve.

The Addition Property of Equality When solving an equation, we have to apply the same operation to each side of the equa- tion. For example, one way to solve the equation

x + 2 = 5

is to add -2 to each side. This step results in isolating the x on one side of the equation.

x + 2 = 5 Given equation x + 2 � (�2) = 5 � (�2) Add -2 to each side. x + 0 = 3 Addition of real numbers x = 3 Additive identity

MAKING CONNECTIONS

Equations and Expressions

Although the words “equation” and “expression” occur frequently in mathematics, they are not interchangeable. An equation always contains an equals sign but an expression never contains an equals sign. We often want to solve an equation, whereas an expression can sometimes be simplified. Furthermore, the equals sign in an equation separates two expres- sions. For example, 3x - 5 = x + 1 is an equation where 3x - 5 and x + 1 are each expressions.

READING CHECK

• What is the difference between an equation and an expression?

STUDY TIP

The addition property of equality is used to solve equations throughout the remainder of the text. Be sure that you have a firm under- standing of this important property.

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

92 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

These four equations (on the bottom of the previous page) are equivalent, but the solution is easiest to see in the last equation. When �2 is added to each side of the given equation, the addition property of equality is used.

READING CHECK

• Why do we use the addi- tion property of equality?

ADDITION PROPERTY OF EQUALITY

If a, b, and c are real numbers, then

a = b is equivalent to a + c = b + c.

That is, adding the same number to each side of an equation results in an equivalent equation.

NOTE: Because any subtraction problem can be changed to an addition problem, the addi- tion property of equality also works for subtraction. That is, if the same number is sub- tracted from each side of an equation, the result is an equivalent equation.

EXAMPLE 1 Using the addition property of equality

Solve each equation. (a) x + 10 = 7 (b) t - 4 = 3 (c) 12 = -

3 4 + y

Solution (a) When solving an equation, we try to isolate the variable on one side of the equation. If

we add -10 to (or subtract 10 from) each side of the equation, we find the value of x.

x + 10 = 7 Given equation x + 10 � 10 = 7 � 10 Subtract 10 from each side. x + 0 = -3 Addition of real numbers x = -3 Additive identity

The solution is -3. (b) To isolate the variable t, add 4 to each side.

t - 4 = 3 Given equation t - 4 � 4 = 3 � 4 Add 4 to each side. t + 0 = 7 Addition of real numbers t = 7 Additive identity

The solution is 7. (c) To isolate the variable y, add 34 to each side.

1

2 = -

3

4 + y Given equation

1

2 �

3 4

= - 3

4 �

3 4

+ y Add 34 to each side.

5

4 = 0 + y Addition of real numbers

5

4 = y Additive identity

The solution is 54.

Now Try Exercises 17, 19, 23

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932.1 INTRODUCTION TO EQUATIONS

CHECKING A SOLUTION To check a solution, substitute it in the given equation to find out if a true statement results. To check the solution for Example 1(c), substitute 54 for y in the given equation. Note that a question mark is placed over the equals sign when a solution is being checked.

1

2 = -

3

4 + y Given equation

1

2 � -

3

4 +

5 4

Replace y with 54.

1

2 � 2

4 Add fractions.

1

2 =

1

2 ✓ The answer checks.

The answer of 54 checks because the resulting equation is true.

CRITICAL THINKING

When you are checking a solution, why do you substi- tute your answer in the given equation?

EXAMPLE 2 Solving an equation and checking a solution

Solve the equation -5 + y = 3 and then check the solution.

Solution Isolate y by adding 5 to each side.

-5 + y = 3 Given equation -5 � 5 + y = 3 � 5 Add 5 to each side. 0 + y = 8 Addition of real numbers y = 8 Additive identity

The solution is 8. To check this solution, substitute 8 for y in the given equation.

-5 + y = 3 Given equation -5 + 8 � 3 Replace y with 8. 3 = 3 ✓ Add; the answer checks.

Now Try Exercise 21

MAKING CONNECTIONS

Equations and Scales

Think of an equation as an old-fashioned scale, where two pans must balance, as shown in the figure. If the two identical golden weights balance the pans, then adding identical red weights to each pan results in the pans remaining balanced. Similarly, removing (subtract- ing) identical weights from each side will also keep the pans balanced.

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94 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

The Multiplication Property of Equality We can illustrate the multiplication property of equality by considering a formula that con- verts yards to feet. Because there are 3 feet in 1 yard, the formula F = 3Y computes F, the number of feet in Y yards. For example, if Y = 5 yards, then F = 3 # 5 = 15 feet.

Now consider the reverse, converting 27 feet to yards. The answer to this conversion corresponds to the solution to the equation

27 = 3Y.

To find the solution, multiply each side of the equation by the reciprocal of 3, or 13.

27 = 3Y Given equation

1 3 # 27 = 1

3 # 3 # Y Multiply each side by 13.

9 = 1 # Y Multiplication of real numbers 9 = Y Multiplicative identity

Thus 27 feet are equivalent to 9 yards.

STUDY TIP

The multiplication property of equality is used to solve equations throughout the remainder of the text. Be sure that you have a firm under- standing of this important property.

READING CHECK

• Why do we use the multiplication property of equality?

MULTIPLICATION PROPERTY OF EQUALITY

If a, b, and c are real numbers with c � 0, then

a = b is equivalent to ac = bc.

That is, multiplying each side of an equation by the same nonzero number results in an equivalent equation.

NOTE: Because any division problem can be changed to a multiplication problem, the mul- tiplication property of equality also works for division. That is, if each side of an equation is divided by the same nonzero number, the result is an equivalent equation.

EXAMPLE 3 Using the multiplication property of equality

Solve each equation. (a) 13 x = 4 (b) -4y = 8 (c) 5 =

3 4 �

Solution (a) We start by multiplying each side of the equation by 3, the reciprocal of 13.

1

3 x = 4 Given equation

3 # 1 3

x = 3 # 4 Multiply each side by 3. 1 # x = 12 Multiplication of real numbers x = 12 Multiplicative identity

The solution is 12.

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952.1 INTRODUCTION TO EQUATIONS

(b) The coefficient of the y-term is -4, so we can either multiply each side of the equation by - 14 or divide each side by �4. This step will make the coefficient of y equal to 1.

-4y = 8 Given equation

-4y �4

= 8

�4 Divide each side by -4.

y = -2 Simplify fractions.

The solution is -2. (c) To change the coefficient of � from 34 to 1, multiply each side of the equation by

4 3, the

reciprocal of 34.

5 = 3

4 � Given equation

4 3 # 5 = 4

3 # 3

4 � Multiply each side by 43.

20

3 = 1 # � Multiplication of real numbers

20

3 = � Multiplicative identity

The solution is 203 .

Now Try Exercises 37, 43, 45

EXAMPLE 4 Solving an equation and checking a solution

Solve the equation 34 = - 3 7 t and then check the solution.

Solution Multiply each side of the equation by �73, the reciprocal of -

3 7.

3

4 = -

3

7 t Given equation

� 7 3 # 3

4 = �

7 3 # a - 3

7 b t Multiply each side by - 73.

- 7

4 = 1 # t Multiplication of real numbers

- 7

4 = t Multiplicative identity

The solution is - 74. To check this answer, substitute � 7 4 for t in the given equation.

3

4 = -

3

7 t Given equation

3

4 � -

3

7 # a� 7

4 b Replace t with - 74.

3

4 =

3

4 ✓ Multiply; the answer checks.

Now Try Exercise 47

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

96 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

N REAL-WORLD CONNECTION Twitter is a microblogging Web site that is used to post short messages called “tweets” on the Internet. In its early years, Twitter’s popularity increased dramatically and new accounts were added at an amazing rate. People from around the world began posting millions of tweets every day. (Source: Twitter.)

EXAMPLE 5 Analyzing Twitter account data

In the early months of 2010, Twitter added 0.3 million new accounts every day. (a) Write a formula that gives the number of new Twitter accounts T added in x days. (b) At this rate, how many days would be needed to add 18 million new accounts?

Solution (a) In 1 day 0.3 # 1 = 0.3 million new accounts were added, in 2 days 0.3 # 2 = 0.6 mil-

lion new accounts were added, and in x days 0.3 # x = 0.3x new accounts were added. So the formula is T = 0.3 x, where x is in days and T is in millions.

(b) To find the number of days needed for Twitter to add 18 million new accounts, replace the variable T in the formula with 18 and solve the resulting equation.

T = 0.3x Formula from part (a) 18 = 0.3x Replace T with 18.

18

0.3 =

0.3x

0.3 Divide each side by 0.3.

60 = x Simplify.

At this rate, it takes 60 days to add 18 million accounts.

Now Try Exercise 59

2.1 Putting It All Together

Equation An equation is a mathematical state- ment that two expressions are equal. An equation can be either true or false.

The equation 2 + 3 = 5 is true. The equation 1 + 3 = 7 is false.

Solution A value for a variable that makes an equation a true statement

The solution to x + 5 = 20 is 15, and the solutions to x2 = 9 are -3 and 3.

Solution Set The set of all solutions to an equation The solution set to x + 5 = 20 is {15}, and the solution set to x2 = 9 is{-3, 3}.

CONCEPT COMMENTS EXAMPLES

Equivalent Equations Two equations are equivalent if they have the same solution set.

The equations

2 x = 14 and x = 7

are equivalent because the solution set to both equations is {7}.

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CONCEPT COMMENTS EXAMPLES

Addition Property of Equality

The equations

a = b and a + c = b + c

are equivalent. This property is used to solve equations.

To solve x - 3 = 8, add 3 to each side of the equation.

x - 3 + 3 = 8 + 3 x = 11

The solution is 11.

Multiplication Property of Equality

When c � 0, the equations

a = b and a # c = b # c are equivalent. This property is used to solve equations.

To solve 15 x = 10, multiply each side of the equation by 5.

5 # 1 5

x = 5 # 10 x = 50

The solution is 50.

Checking a Solution Substitute the solution for the variable in the given equation and then sim- plify each side to see if a true state- ment results.

To show that 8 is a solution to

x + 12 = 20,

substitute 8 for x.

8 + 12 � 20 20 = 20 True

972.1 INTRODUCTION TO EQUATIONS

2.1 Exercises

CONCEPTS AND VOCABULARY

1. Each value of a variable that makes an equation true is called a(n) _____.

2. The equation 1 + 3 = 4 is (true/false).

3. The equation 2 + 3 = 6 is (true/false).

4. The _____ is the set of all solutions to an equation.

5. To solve an equation, find all _____.

6. Equations with the same solution set are called _____ equations.

7. If a = b, then a + c = _____.

8. Because any subtraction problem can be changed to an addition problem, the addition property of equality also works for _____.

9. If a = b and c � 0, then ac = _____ .

10. Because any division problem can be changed to a multiplication problem, the multiplication property of equality also works for _____.

11. To solve an equation, transform the equation into a(n) _____ equation that is easier to solve.

12. To check a solution, substitute it for the variable in the _____ equation.

THE ADDITION PROPERTY OF EQUALITY

13. To solve x - 22 = 4, add _____ to each side.

14. To solve 56 = 1 6 + x, add _____ to each side.

15. To solve x + 3 = 13, subtract _____ from each side.

16. To solve 34 = 1 4 + x, subtract _____ from each side.

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98 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

Exercises 17–30: Solve the equation. Check your answer.

17. x + 5 = 0 18. x + 3 = 7

19. a - 12 = -3 20. a - 19 = -11

21. 9 = y - 8 22. 97 = -23 + y

23. 15 = z - 3 2 24.

3 4 + z = -

1 2

25. t - 0.8 = 4.3 26. y - 1.23 = -0.02 27. 4 + x = 1 28. 16 + x = -2

29. 1 = 13 + y 30. 7 2 = -2 + y

31. Thinking Generally To solve x - a = b for x, add _____ to each side.

32. Thinking Generally To solve x + a = b for x, subtract _____ from each side.

THE MULTIPLICATION PROPERTY OF EQUALITY

33. To solve 5x = 4, multiply each side by _____.

(c) Use your formula to calculate the total rainfall at 3 P.M. Does the answer agree with the value in your table from part (a)?

(d) How much rain has fallen by 2:15 P.M.?

56. Cold Weather A furnace is turned on at midnight when the temperature inside a cabin is 0� F. The cabin warms at a rate of 10� F per hour until 7 A.M. (a) Make a table that shows the cabin temperature

T in degrees Fahrenheit, x hours past midnight, ending at 7 A.M.

(b) Write a formula that calculates T. (c) Use your formula to calculate the temperature at

5 A.M. Does the answer agree with the value in your table from part (a)?

(d) Find the cabin temperature at 2:45 A.M.

57. Football Field A football field is 300 feet long. (a) Write a formula that gives the length L in feet of

x football fields. (b) Use your formula to write an equation whose solu-

tion gives the number of football fields in 870 feet. (c) Solve your equation from part (b).

58. Acreage An acre equals 43,560 square feet. (a) Write a formula that converts A acres to S square

feet. (b) Use your formula to write an equation whose

solution gives the number of acres in 871,200 square feet.

(c) Solve your equation from part (b).

59. Twitter Accounts (Refer to Example 5.) In the early months of 2010, Twitter added 0.3 million new accounts every day. At this rate, how many days would be needed to add 15 million new Twitter accounts? (Source: Twitter.)

60. Web Site Visits If a Web site was increasing its num- ber of visitors by 14,000 every day, how many days would it take for the site to gain a total of 98,000 new visitors?

61. Online Exploration The city of Winnipeg is located in the province of Manitoba in Canada. (a) Use the Internet to find the latitude of Winnipeg

to the nearest degree. (b) Use the table on page 89 to determine how many

times as intense the sun’s ultraviolet rays are at the equator (latitude 0°) on March 21 compared to the sun’s intensity in Winnipeg. Round your answer to 1 decimal place.

62. Online Exploration Columbus is a city located in the center of Ohio. It is the state’s capital city. (a) Use the Internet to find the latitude of Columbus

to the nearest degree.

34. To solve 43 y = 8, multiply each side by _____.

35. To solve 6 x = 11, divide each side by _____.

36. To solve 0.2 x = 4, divide each side by _____.

Exercises 37–52: Solve the equation. Check your answer.

37. 5x = 15 38. -2 x = 8

39. -7x = 0 40. 25x = 0

41. -35 = -5a 42. -32 = -4a

43. -18 = 3a 44. -70 = 10a

45. 12 x = 3 2 46.

3 4 x =

5 8

47. 12 = 2 5 z 48. -

3 4 = -

1 8 z

49. 0.5t = 3.5 50. 2.2t = -9.9

51. -1.7 = 0.2 x 52. 6.4 = 1.6 x

53. Thinking Generally To solve 1a # x = b for x, multi- ply each side by _____ .

54. Thinking Generally To solve ax = b for x, where a � 0, divide each side by _____ .

APPLICATIONS

55. Rainfall On a stormy day it rains 3 inches before noon and 12 inch per hour thereafter until 6 P.M. (a) Make a table that shows the total rainfall R in

inches, x hours past noon, ending at 6 P.M. (b) Write a formula that calculates R.

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992.2 LINEAR EQUATIONS

(b) Use the table on page 89 to determine how many times as intense the sun’s ultraviolet rays are in Limon, Costa Rica, (latitude 10°N) on June 21 compared to the sun’s intensity in Columbus. Round your answer to 1 decimal place.

63. Cost of a Car When the cost of a car is multiplied by 0.07 the result is $1750. Find the cost of the car.

64. Raise in Salary If an employee’s salary is multiplied by 1.06, which corresponds to a 6% raise, the result is $58,300. Find the employee’s current salary.

WRITING ABOUT MATHEMATICS

65. A student solves an equation as follows.

x + 30 = 64 x � 64 + 30 x � 94

Identify the student’s mistake. What is the solution?

66. What is a good first step for solving the equation a b x = 1, where a and b are natural numbers? What is the solution? Explain your answers.

2.2 Linear Equations Basic Concepts ● Solving Linear Equations ● Applying the Distributive Property ● Clearing Fractions and Decimals ● Equations with No Solutions or Infinitely Many Solutions

A LOOK INTO MATH N Billions of dollars are spent each year to solve equations that lead to the creation of better products. If our society could not solve equations, we would not have HDTV, high-speed Internet, satellites, fiber optics, CAT scans, smart phones, or accurate weather forecasts. In this section we discuss linear equations and some of their applications. Linear equations can always be solved by hand.

Basic Concepts Suppose that a bicyclist is 5 miles from home, riding away from home at 10 miles per hour, as shown in Figure 2.1. The distance between the bicyclist and home for various elapsed times is shown in Table 2.3.NEW VOCABULARY

n Linear equation n Identity n Contradiction

TABLE 2.3 Distance from Home

Elapsed Time (hours) 0 1 2 3

Distance (miles) 5 15 25 35

10 miles 10 miles 10 miles

The bicyclist is moving at a constant speed, so the distance increases by 10 miles every hour. The distance D from home after x hours can be calculated by the formula

D = 10 x + 5.

For example, after 2 hours the distance is

D = 10(2) + 5 = 25 miles.

Table 2.3 verifies that the bicyclist is 25 miles from home after 2 hours. However, the table is less helpful if we want to find the elapsed time when the bicyclist is 18 miles from home. To answer this question, we could begin by substituting 18 for D in the formula to obtain the equation

18 = 10 x + 5.

Figure 2.1 Distance from Home

5 mi

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100 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

The equation 18 = 10x + 5 can be written in a different form by applying the addition property of equality. Subtracting 18 from each side gives an equivalent equation.

18 � 18 = 10x + 5 � 18 Subtract 18 from each side. 0 = 10x - 13 Simplify. 10 x - 13 = 0 Rewrite the equation.

Even though these steps did not result in a solution to the equation 18 = 10x + 5, apply- ing the addition property of equality allowed us to rewrite the equation as 10 x - 13 = 0, which is an example of a linear equation. (See Example 3(a) for a solution to the equation 10x - 13 = 0.) Linear equations can model applications in which things move or change at a constant rate.

LINEAR EQUATION IN ONE VARIABLE

A linear equation in one variable is an equation that can be written in the form

a x + b = 0,

where a and b are constants with a � 0.

If an equation is linear, writing it in the form a x + b = 0 should not require any prop- erties or processes other than the following.

• using the distributive property to clear any parentheses • combining like terms • applying the addition property of equality

For example, the equation 18 = 10 x + 5 is linear because applying the addition property of equality results in 10 x - 13 = 0, as shown above.

Table 2.4 gives examples of linear equations and values for a and b.

NOTE: An equation cannot be written in the form a x + b = 0 if after clearing parentheses and combining like terms, any of the following statements are true.

1. The variable has an exponent other than 1. 2. The variable appears in a denominator of a fraction. 3. The variable appears under the symbol 1 or within an absolute value.

TABLE 2.4 Linear Equations

Equation In a x + b = 0 Form a b

x = 1 x - 1 = 0 1 -1

-5x + 4 = 3 -5x + 1 = 0 -5 1

2.5x = 0 2.5x + 0 = 0 2.5 0

READING CHECK

• Name three things that tell you that an equation is not a linear equation.

EXAMPLE 1 Determining whether an equation is linear

Determine whether the equation is linear. If the equation is linear, give values for a and b that result when the equation is written in the form ax + b = 0. (a) 4 x + 5 = 0 (b) 5 = - 34 x (c) 4x

2 + 6 = 0 (d) 3x + 5 = 0

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1012.2 LINEAR EQUATIONS

Solution (a) The equation is linear because it is in the form a x + b = 0 with a = 4 and b = 5. (b) The equation can be rewritten as follows.

5 = - 3

4 x Given equation

3 4

x + 5 = 3 4

x + a - 3 4

xb Add 34 x to each side.

3

4 x + 5 = 0 Additive inverse

The given equation is linear because it can be written in the form a x + b = 0 with a = 34 and b = 5.

NOTE: If 5 had been subtracted from each side, the result would be 0 = - 34 x - 5, which is an equivalent linear equation with a = - 34 and b = -5.

(c) The equation is not linear because it cannot be written in the form a x + b = 0. The variable has exponent 2.

(d) The equation is not linear because it cannot be written in the form a x + b = 0. The variable appears in the denominator of a fraction.

Now Try Exercises 9, 11, 13, 15

Solving Linear Equations Every linear equation has exactly one solution. Showing that this is true is left as an exercise (see Exercise 59). Solving a linear equation means finding the value of the variable that makes the equation true.

SOLVING LINEAR EQUATIONS NUMERICALLY One way to solve a linear equation is to make a table of values. A table provides an organized way of checking possible values of the variable to see if there is a value that makes the equation true. For example, if we want to solve the equation

2 x - 5 = �7,

we substitute various values for x in the left side of the equation. If one of these values results in �7, then the value makes the equation true and is the solution. In the next exam- ple a table of values is used to solve this equation.

EXAMPLE 2 Using a table to solve an equation

Complete Table 2.5 for the given values of x. Then solve the equation 2 x - 5 = -7.

TABLE 2.5

x -3 -2 -1 0 1 2 3

2 x - 5 -11

Solution To complete the table, substitute x = -2, -1, 0, 1, 2, and 3 into the expression 2 x - 5. For example, if x = -2, then 2 x - 5 = 2(�2) - 5 = -9. The other values shown in Table 2.6 on the next page can be found similarly.

IS B

N 1-

25 6-

49 08

2- 2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

102 CHAPTER 2 LINEAR EQUATIONS AND INEQUALITIES

From the table, 2 x - 5 equals �7 when x = �1. So the solution to 2 x - 5 = -7 is -1.

Now Try Exercise 25

TABLE 2.6

x -3 -2 �1 0 1 2 3

2 x - 5 -11 -9 �7 -5 -3 -1 1

SOLVING LINEAR EQUATIONS SYMBOLICALLY Although tables can be used to solve some linear equations, the process of creating a table that contains the solution can take a significant amount of time. For example, the solution to the equation 9x - 4 = 0 is 4 9. However, creating a table that reveals this solution would be quite challenging.

The following strategy, which involves the addition and multiplication properties of equality, is a method for solving linear equations symbolically.

TECHNOLOGY NOTE

Graphing Calculators and Tables Many graphing calculators have the capability to make tables. Table 2.6 is shown in the accompanying figure.

Y1�2X�5

X Y1 �3 �11 �2 �9 �1 �7 0 �5 1 �3 2 �1 3 1

CALCULATOR HELP To make a table on a calculator, see Appendix A (pages AP-2 and AP-3). SOLVING A LINEAR EQUATION SYMBOLICALLY

STEP 1: Use the distributive property to clear any parentheses on each side of the equation. Combine any like terms on each side.