# mathemathics

11

875

Systems of Equations and Inequalities

Figure 1 enigma machines like this one, once owned by Italian dictator benito mussolini, were used by government and military officials for enciphering and deciphering top-secret communications during World War II. (credit: dave Addey, Flickr)

Introduction By 1943, it was obvious to the Nazi regime that defeat was imminent unless it could build a weapon with unlimited destructive power, one that had never been seen before in the history of the world. In September, Adolf Hitler ordered German scientists to begin building an atomic bomb. Rumors and whispers began to spread from across the ocean. Refugees and diplomats told of the experiments happening in Norway. However, Franklin D. Roosevelt wasn’t sold, and even doubted British Prime Minister Winston Churchill’s warning. Roosevelt wanted undeniable proof. Fortunately, he soon received the proof he wanted when a group of mathematicians cracked the “Enigma” code, proving beyond a doubt that Hitler was building an atomic bomb. The next day, Roosevelt gave the order that the United States begin work on the same.

The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.

Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.

In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.

ChAPTeR OUTlIne

11.1 Systems of linear equations: Two variables 11.2 Systems of linear equations: Three variables 11.3 Systems of nonlinear equations and Inequalities: Two variables 11.4 Partial Fractions 11.5 matrices and matrix Operations 11.6 Solving Systems with gaussian elimination 11.7 Solving Systems with Inverses 11.8 Solving Systems with Cramer’s Rule

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

SECTION 11.1 sectioN exercises 889

11.1 SeCTIOn exeRCISeS

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1. Can a system of linear equations have exactly two solutions? Explain why or why not.

2. If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.

3. If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

4. If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

5. Given a system of equations, explain at least two different methods of solving that system.

AlgebRAIC

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

6. 5x − y = 4 x + 6y = 2 and (4, 0)

7. −3x − 5y = 13 − x + 4y = 10 and (−6, 1)

8. 3x + 7y = 1 2x + 4y = 0 and (2, 3)

9. −2x + 5y = 7 2x + 9y = 7 and (−1, 1)

10. x + 8y = 43 3x − 2y = −1 and (3, 5)

For the following exercises, solve each system by substitution.

11. x + 3y = 5 2x + 3y = 4

12. 3x − 2y = 18 5x + 10y = −10

13. 4x + 2y = −10 3x + 9y = 0

14. 2x + 4y = −3.8 9x − 5y = 1.3

15. −2x + 3y = 1.2 −3x − 6y = 1.8

16. x − 0.2y = 1 −10x + 2y = 5

17. 3 x + 5y = 9 30x + 50y = −90

18. −3x + y = 2 12x − 4y = −8

19. 1 __ 2 x + 1 __ 3 y = 16

1 __ 6 x + 1 __ 4 y = 9

20. − 1 __ 4 x + 3 __ 2 y = 11

− 1 __ 8 x + 1 __ 3 y = 3

For the following exercises, solve each system by addition.

21. −2x + 5y = −42 7x + 2y = 30

22. 6x − 5y = −34 2x + 6y = 4

23. 5x − y = −2.6 −4x − 6y = 1.4

24. 7x − 2y = 3 4x + 5y = 3.25

25. −x + 2y = −1 5x − 10y = 6

26. 7x + 6y = 2 −28x − 24y = −8

27. 5 __ 6 x + 1 __ 4 y = 0

1 __ 8 x − 1 __ 2 y = −

43 ___ 120

28. 1 __ 3 x + 1 __ 9 y =

2 __ 9

− 1 __ 2 x + 4 __ 5 y = −

1 __ 3

29. −0.2x + 0.4y = 0.6 x − 2y = −3

30. −0.1x + 0.2y = 0.6 5x − 10y = 1

For the following exercises, solve each system by any method.

31. 5x + 9y = 16 x + 2y = 4

32. 6x − 8y = −0.6 3x + 2y = 0.9

33. 5x − 2y = 2.25 7x − 4y = 3

34. x − 5 ___ 12 y = − 55 ___ 12

−6x + 5 __ 2 y = 55 ___ 2

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

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890 CHAPTER 11 systems of equatioNs aNd iNequalities

35. 7x − 4y = 7 __ 6

2x + 4y = 1 __ 3

36. 3x + 6y = 11 2x + 4y = 9

37. 7 __ 3 x − 1 __ 6 y = 2

− 21 ___ 6 x + 3 ___ 12 y = −3

38. 1 __ 2 x + 1 __ 3 y =

1 __ 3

3 __ 2 x + 1 __ 4 y = −

1 __ 8

39. 2.2x + 1.3y = −0.1 4.2x + 4.2y = 2.1

40. 0.1x + 0.2y = 2 0.35x − 0.3y = 0

gRAPhICAl

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

41. 3x − y = 0.6 x − 2y = 1.3

42. −x + 2y = 4 2x − 4y = 1

43. x + 2y = 7 2x + 6y = 12

44. 3x − 5y = 7 x − 2y = 3

45. 3x − 2y = 5 −9x + 6y = −15

TeChnOlOgy

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

46. 0.1x + 0.2y = 0.3 −0.3x + 0.5y = 1

47. −0.01x + 0.12y = 0.62 0.15x + 0.20y = 0.52

48. 0.5x + 0.3y = 4 0.25x − 0.9y = 0.46

49. 0.15x + 0.27y = 0.39 −0.34x + 0.56y = 1.8

50. −0.71x + 0.92y = 0.13 0.83x + 0.05y = 2.1

exTenSIOnS

For the following exercises, solve each system in terms of A, B, C, D, E, and F where A – F are nonzero numbers. Note that A ≠ B and AE ≠ BD.

51. x + y = A x − y = B

52. x + Ay = 1 x + By = 1

53. Ax + y = 0 Bx + y = 1

54. Ax + By = C x + y = 1

55. Ax + By = C Dx + Ey = F

ReAl-WORld APPlICATIOnS

For the following exercises, solve for the desired quantity.

56. A stuffed animal business has a total cost of production C = 12x + 30 and a revenue function R = 20x. Find the break-even point.

57. A fast-food restaurant has a cost of production C(x) = 11x + 120 and a revenue function R(x) = 5x. When does the company start to turn a profit?

58. A cell phone factory has a cost of production C(x) = 150x + 10,000 and a revenue function R(x) = 200x. What is the break-even point?

59. A musician charges C(x) = 64x + 20,000, where x is the total number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

60. A guitar factory has a cost of production C(x) = 75x + 50,000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

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SECTION 11.1 sectioN exercises 891

For the following exercises, use a system of linear equations with two variables and two equations to solve.

61. Find two numbers whose sum is 28 and difference is 13.

62. A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.

63. The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?

64. A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

65. A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?

66. 276 students enrolled in a freshman-level chemistry class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.

67. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?

68. A jeep and BMW enter a highway running east- west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

69. If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?

70. An investor earned triple the profits of what she earned last year. If she made $500,000.48 total for both years, how much did she earn in profits each year?

71. An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110% increase on the money she invested. On the second investment, Riverside Community, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?

72. If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the

other that pays 2 7 __ 8 % interest, and the investor

earns $737.50 annual interest, how much was invested in each account?

73. If an investor invests $23,000 into two bonds, one that pays 4% in simple…

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