Download

(1)

7.1 Graphing Systems of Equations

1. (1, 5)

2. (–4, 6)

3. ∅

4. y = 2x – 3

5. yes

6. no

7. no

8. yes

9. no

10. yes

11. no

12. yes

13. yes

14. no

15.

16.

y

x

(3, 2)

y

(2)

Section 7.1 y

x

(1, 1)

17.

18.

19.

20.

y

x (22, 22)

y

x

(3, ₂5)

y

x

(3)

Section 7.1

21.

22.

23.

24.

y

x

(₂2, 6)

y

x

(22, 0)

y

x (5, 7)

y

x

(4)

Section 7.1

y

x (2, 4)

y

x

(₂2, ₂3)

y

x

(₂3, ₂1)

(7, ₂1) y

x

25.

26.

27.

(5)

Section 7.1

29. ∅

y

x

30. y = –

2

x

+ 4

y

x

31. ∅

y

x

32. y = x – 6

y

(6)

Section 7.1

33a. x + y = –7 and x – y = 3

33b.

34a. cM(x) = 5x + 45; cJ(x) = 14x

34b. 5 people for $70

(5, 70)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

0 5

People

Doll

ars

34c. when x > 5 or for at least 6

people

y

x

(7)

Section 7.1

35a. r(x) = 30x

35b. e(x) = 20x + 120

35c. 12 jackets

0 50 100 150 200 250 300 350 400 450 500

0 10 20 30

Jackets

Jacket Sales

D

oll

ars

36.

y

x

(8)

Section 7.1

37. about 230 min

0 100 200 300 400 500

Minutes

0 10 20 30 40 50 60 70 80 90 100

Cost of Cell Phone Plans

Dollars plan B

plan A

38.

Year Mexico (in

millions)

Colombia (in

millions)

1950 26 12

1960 35 16

1970 52 21

1980 69 27

1990 83 33

2000 100 40

(9)

Section 7.1

0

1940 1960 1980 2000 2020 20

40 60 80 100 120

Mexican and Colombian Populations

M

illions of People

Year

Mexico Colombia 39.

40. 37

15

41. 7

78

42. 27

8

43. x = 5

44. x = 29

5

45. x = 4 5

3

y

− +

46. y = –

3

x

+ 7 or y = –1

3x + 7

47. x ≥

–3

48. x = 4

(10)

7.2 Solving Simple Systems by Substitution

1. a

2. d

3. c

4. b

5. (7, 21)

6. (–8, –32)

7. (4, 14)

8. (–1, 10)

9. (8, 3)

10. (4, 1)

11. (–2, –6)

12. (–1, 12)

13. (–4, 3)

14. (5, –7)

15. (9, 2)

16. (1, 10)

17. (–2, 8)

18. (47, –19)

19. (–3, –4)

20. (2, 7)

21. (–2, 1)

(11)

Section 7.2

23. (–4, 5)

24. (10, –6)

25. (10, 5)

26. 2 2

5 3,

⎛ ⎞

⎜ ⎟

⎝ ⎠

27. 28 14

5 , 5

⎛ ⎞

⎜ ⎟

⎝ ⎠

28. 10 8

3 3

– , –

⎛ ⎞

⎜ ⎟

⎝ ⎠

29. 38 and 25

30. 675 and 321

31. 19 ft × 26 ft

32. 3 sheets

33. 5 apple and 7 pecan

34. (–1, –6)

35. 66 26

41 41,

⎛ ⎞

⎜ ⎟

(12)

Section 7.2

36. Mexico: y = 1.49x – 2874

Colombia: y = 0.58x – 1118

0

1940 1960 1980 2000 2020

20 40 60 80 100 120

Millions of People

Mexican and Colombian Populations

Year

Mexico Colombia

37. No; the Mexican population is larger and is growing at a faster rate.

38. c

39. b

40. 7y + 14

41. x + 7

42. x2 + 2x + 1

43. x2 + 2x + 1

44. 8x + 17

45. 2x + 14

46. 10

(13)

7.3 Solving Advanced Systems by Substitution

1. c

2. b

3. d

4. a

5. a

6. c

7. b

8. a

9. d

10. b and d

11. (–8, –1)

12. (0, 6)

13. (9, –1)

14. (3, –1)

15. (9, –1)

16. (1, –1)

17. 3 1

5

,

⎛ ⎞

⎜ ⎟

⎝ ⎠

18. (3, 11)

19. 9 25

26, 26

⎛ ⎞

⎜ ⎟

⎝− − ⎠

(14)

Section 7.3

21. (–4, 3)

22. (–3, 6)

23. 5 4

2 3,

⎛ ⎞

⎜ ⎟

⎝ ⎠

24. 1 7

6 10,

⎛ ⎞

⎜ ⎟

⎝ ⎠

25. 1 5

3 6,

⎛ ⎞

⎜ ⎟

⎝ ⎠

26. 1 3

10, 8

⎛ ⎞

⎜ ⎟

⎝ − ⎠

27. 63 15

23, 23

⎛ ⎞

⎜ ⎟

⎝ − ⎠

28. 67 5

22, 22

⎛ ⎞

⎜ ⎟

⎝ − ⎠

29. wallet: $0.50; 5” × 7”: $5

30. down payment: $800; bank loan: $4000

31. 4% investment: ≈ 6.08 yr

(6 yr 1 mo); 5% investment

≈ 7.08 yr (7 yr 1 mo)

32. peach: $4.50; cherry: $5.75

33. 4 cheesecakes and 5 pies

34. 1 3

2 4,

⎛ ⎞

⎜ ⎟

(15)

Section 7.3

35. 3 6

2 5,

⎛ ⎞

⎜ ⎟

⎝− ⎠ or (–1.5, 1.2)

36. (1, 4) and (–1, 4)

37. CD: 16 yr;

mutual fund: 15 yr

38. CD Creators:

c(x) = 0.18x + 50;

Digital Dzign: c(x) = 0.2x

39. 2500 CDs

40. CD Creators; $1850

41. x =3, 5

42. x = –8

43. x = –10

3 , 6

44. x < 3 or x > 5

45. ∅

46. x ≠ –8

47. f(x) = 2x + 1

48. g (x) = 3x + 2

49. m = –1

(16)

7.4 Solving Systems by Elimination

1a. 2nd equation by –7

1b. either equation by –1

2a. 1st equation by –9

2b. just add

3a. 2nd equations by –2

3b. either equation b –1

4a. either equation by –1

4b. 2nd equation by 2

5. (1, 4)

6. (10, –11)

7. (–3, –1)

8. (–3, 8)

9. (–3, 4)

10. (–1, 7)

11–14. Answers given assume equations will be added.

11b. 1st equation by 5 and

2nd equation by 3

(17)

Section 7.4

2nd equation by 11

2nd equation by 2

15. , 5

2

4

⎛ ⎞

⎜ ⎟

⎝ ⎠

16. 1

3

9, –

⎛ ⎞

⎜ ⎟

⎝ ⎠

17. (4, 7)

18. (4, 3)

19. (–1, –3)

20. (–8, –2)

21. (6, –6)

22. (–5, –7)

23. 5 9

2 2,

⎛ ⎞

⎜ ⎟

⎝ ⎠

24. 23 11

5 , 5

⎛ ⎞

⎜ ⎟

⎝ ⎠

25. 19 1

8 , 16

⎛ ⎞

⎜ ⎟

⎝ ⎠

26. 77 79

23 23,

⎛ ⎞

⎜ ⎟

⎝ ⎠

27. 31 4

17 17,

⎛ ⎞

⎜ ⎟

(18)

Section 7.4

28. (10, 4)

29. 365 and 256

30. 98 and 36

31. 75 adult and

170 student tickets

32. 25 true/false and 25 short answer

33. 18 softballs and 25 baseballs

34. 24

7 and

8 7

35. 2 oz: 23; 3 oz: 14

36. 41 67

18 30,

⎛ ⎞

⎜ ⎟

⎝ ⎠

37. (–8, 25)

38a. Express Press: $490;

Copy Center: $550

38b. $777.50

39. 200 CDs per person

40. 50 people

41. 10

42. –5

43. –2x2 + x + 6

(19)

Section 7.4

45. x = 14b c

a

+

46. x =

10

a c

47. 43.75%

48. 40

49. 31.5 L

(20)

7.5 Special Systems

1. b

2. c

3. b

4. c

5. a, d

6. a, b

7. c

8. a, d

9. a, b

10. b

11. c

12. c

13. b

14. a

15. (–3, –4);

consistent independent

16. (4, 2);

17. ∅; inconsistent

18. y = –x + 7;

consistent dependent

19. (2, 4);

(21)

Section 7.5

20. (–2, 6);

21. y = –4

3x + 7 3;

24. (3, –5);

25. 41, – 3

7 7

⎛ ⎞

⎜ ⎟

⎝ ⎠ ;

26. y = –x + 10;

27. 63, – 15

23 23

⎛ ⎞

⎜ ⎟

⎝ ⎠ ;

30. y = 5

6x – 7 6;

31. y = 3

2x + 4;

32. They have the same slope

(22)

Section 7.5 0

0 100 200 300

20 40 60 80 100

Easy

Clean

Miles

Dollars

33. They have the same slope

and the same y–intercept.

34. They have different slopes.

Their y–intercepts could be

the same or different.

35. 12 cm, 18 cm, and 18 cm

36. books: $4 each; DVDs: $12 each

37. Easy: y = 0.15x + 50;

Clean: y = 0.2x + 40;

(23)

Section 7.5

38. Each equation has a

fractional y-intercept.

39. Solving for any variable results in a fractional expression.

40. 18 credit hours; if he takes more than 18 hours per semester

41. k = 2; y = 2x

42. k = 4

7; y = 4 7x

43. They both have a y–

intercept of zero or they both go through the origin.

44. 0.445 psi/ft

45. p(d) = 0.445d + 14.7

46. 15,964 psi

47. 1086 atm

48. 1083 psi

(24)

7.6 Motion Problems

1. 12 – w

2. r + 10

3. 4 + c

4. z – 2

5. 500 – w

6a.tb = the time before lunch;

ta = the time after lunch

6b. t_b+ ta = 7

6c. 30t_b+ 40t_a = 230

7a. b = the rate with no wind

7b. 1.125(b + 8) = 36 or

2.25(b – 8) = 36

7c. There is only one unknown

8.

r t d

b + w 2 2(b + w)

(25)

Section 7.6

9.

10.

r t d

150 + w 4 4(150 + w)

150 – w 9 9(150 – w)

11. x = the rate with no wind;

w = the wind speed

12. (x + w) – (x – w) = 10 or

2w = 10

13. 2(x + w) + 5(x – w) = 150 or

7x – 3w = 150

14. bicycles: 23.6 mi/hr; wind: 5mi/hr

15. x = the rate with no wind;

w = the wind speed

16. x = w + 100

17. 2(x + w) = 3(x – w)

18. plane: 125 mi/hr; wind: 25 mi/hr

r t d

b – 3 6 6(b – w)

(26)

Section 7.6

19. 4 hr

20. 4:00 PM

21. plane: 275 knots; wind: 25 knots

22. Mrs. Vaughn: 40 mi/hr; Martin: 52 mi/hr

23. southbound train: 65 mi/hr; northbound train: 55 mi/hr

24. faster car: 60 mi/hr; slower car: 40 mi/hr

25. 24 mi/hr

26. westbound plane: 4 hr; eastbound plane:1 hr

27. first car: 65 mi/hr; second car: 59 mi/hr

28. scouts: 4 mi/hr; current: 1 mi/hr

29. express train: 59.5 mi/hr; local train: 45.5 mi/hr

30. 4 hr

31. 2 hr

32. salmon: 8 mi/hr;

current: 3

4 mi/hr

33. 400,00x + 4300 = 70,000;

x ≈_{$0.164 per CD}

(27)

Section 7.6

35. –32

36. 1

8

37. 81

38. 16

39. 22

40. 37

41. 1₆

3

42. 1₃

2

43.

3

5

2 3

44. ₄ 1₄ ₇

(28)

7.7 Mixture Problems

1. $40

2. $0.65 per apple

3. 82 bags

4. 16d

5. 30x

6. 3 gal of acid

7. 28% salt

8. 150 L

9. 0.12x gal of acid

10. 2p mL of Salt

11. 0%

12. 100%

13. 18 oz of nuts and 30 oz of raisins

14. 6 lb oranges and 4 lb bananas

15. 1.5 lb of caramels and 3.5 lb of butterscotch candles

16. $6000 at 6% and $1500 at 15%

17. 12 gal of the 30% salt solution and

(29)

Section 7.7

18. 8 roses and 6 carnations

19. 40 mL of the 5% solution and 10 mL of the 10% solution

20. 8 g Raisin Rich and 4 g high-protein

21. 35 lb of the original mix and 15 lb of fescue

22. 30 kg of the original concrete mix and 18 kg of aggregate

23. 15% in solution I and 35% in solution II

24. 40 mL of 40% alcohol and 88 mL of water

25. $4200 at 10% and $5800 at 9.5%

26. $1800 at 9.2% and $2500 at 10.4%

27. 116.7 gal of type A and 83.3 gal of type B

28. 27.3 lb of ground beef and 22.7 lb of round steak

29. 10.7 gal cream and 49.3 gal milk

(30)

Section 7.7

24 23 22 21 0 1 2 3 4

31. 26.6 lb of A and 58.4 lb of B

32. $16.77 coffee: 11.4 lb; $4.75 coffee: 38.6 lb

33. dry cleaning: 8%; restaurant: 10%

34. 6426 in district 12 and 3827 in district 13

35. 2.6 L of ocean water and 9.4 L of Dead Sea water

36. corn: $3.30/bu; beans: $9.60/bu

37. 16.5x ≤_2000;_x ≤_121.₂₁_;

121 boxes

38. 4 skids; 100 boxes per skid

39. 1700 lb

40. 8.1%

41. x <_–1

42. y >_–3

(31)

Section 7.7

44. y > _x_{+ 4}

45. y ≤_–1

46. x >₁

47. y

x

y

x

y

x

y

(32)

Section 7.7

y

x

y

x (₂6, 6)

(₂3, 5)

y

x

48.

49. (–6, 6)

(33)

y

x

y

x

7.8 Solving Systems of Inequalities

1. I

2. III

3. II

4. IV

5. yes

6. no

7. no

8. yes

9–14. Points chosen will vary.

9.

(34)

Section 7.8

y

x

y

x

y

x

y

x

11.

12.

13.

(35)

Section 7.8 x

y

x

x y

15. y ≤₂ _and_y ≤ 1

3

− _x +1

16. y <_–_x_{+ 5 and}_y <₂_x_{– 4}

17. y > 1

3x + 4 and y ≤ 2x + 2

18. y > 1

3x – 1 and y < –

1

2x – 4

19.

20.

(36)

Section 7.8 x

y

x

y

x

y

x

22.

23.

24.

(37)

Section 7.8

y

x

y

x

26.

27.

28.

29. y ≥_–3_x_{– 2 and}_y <_–3_x_{+ 5}

30. x >_0,_y >_{4, and}_y ≤_–1

2x + 6

31. y ≥_–3,_x <_{2, and}_y <₂_x_{+ 1}

32. y <_–3_x_{+ 5,}_y >_–3_x_{– 6,}

y < 1

3x + 5, and y

> 1

3x – 2

y

(38)

Section 7.8

0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 10 11

Number of Cakes

Number of Pies

y

x

33a. 12x + 7y ≤_70,

8x + 12y ≥_72,

y ≥_{0, and}_x ≥₀

33b.

33c–d.

Chocolate Silk

Lemon Chiffon

Number of Servings

Cost

2 5 76 $59

2 6 88 $66

(39)

Section 7.8

0 4 8 12 16 20 24

0 2 4 6 8 10 12 14 16 18 y x First Investment ($1000s) Second Investmen t ($1 000s)

34a.

P r t I

x 0.05 5 0.25x

y 0.08 5 0.4y

16,000 5500

34b. x ≥_0,_y ≥_0,_x₊_y ≤_16,000,

and 0.25x + 0.4y ≥₅₅₀₀

34c.

34d. $10,000 at 8% for $5500

interest; $12,000 at 8% for $5800; $14,000 at 8% for $6100

35. any point in the triangular region

0 4 8 12 16 20

(40)

Section 7.8

36. The lines are parallel, and there is no intersection of the shaded regions.

37. 10 small and 25 large (75 hr and $95)

0 10 20 30 40

0 10 20 30 40 50

Lar

ge Dr

esses

Small Dresses

y

x

38. 0.05625x + 6 ≤₂₅

39. 1.5x ≤₁₂₂₄

40. x ≤_{337.78 and}

x ≤_{816; 337 CDs}

41. (11, 19)

42. (–1, 7)

43. (–2, 3)

44. (5, –8)

45. (4, –2)

46. (6, –3)

47. (8, –3)

48. (4, –5)

49. (–4, –2)

(41)

y

x

(₂1, 3)

y

x

(₂1, 2)

Chapter 7 Review

1. yes

2. no

3. yes

4.

5.

6.

y

x

(42)

Chapter 7 Review

7. (4, 0)

8. (–2, –4)

9. ∅

10. (3, 7)

11. (–2, 5)

12. 15 16,

14 7

⎛ ⎞

⎜ ⎟

⎝ ⎠

13. (1, 1)

14. (–3, –4)

15. (2, 4)

16. (6, 9)

17. (–4, 19)

18. y = 2

7x – 3 7

19. (4, 0)

20. (6, –2)

21. ∅

22. 3, – 23

7 7

⎛ ⎞

⎜ ⎟

⎝ ⎠

23. They are parallel.

24. They intersect.

25. They intersect at their

(43)

Chapter 7 Review

26. They coincide.

27. no solutions; inconsistent

28. one solution;

29. infinite number of solutions; consistent dependent

30. one solution;

31. no solutions; inconsistent

32. infinite number of solutions; consistent dependent

33. 12x

34. 3 – c

35. r + 40

36. 0.07d

37. –9 and –14

38. –5 and 7

39. 33 people

40. first car: about 2.04 hr or 2 hr 2 min; second car: 1.79 hr or 1 hr 47 min

41. slower boat: 8 mi/hr; faster boat: 24 mi/hr

(44)

Chapter 7 Review

y

x

43. 2.5 lb of chocolates and 3.5 lb of nuts

44. 3.3 gal of 35% solution and 1.7 gal water

45. 60 gal of 60% sucrose and 240 gal of 25% sucrose

46. $890 at 7.5% and $660 at 8.25%

47.

48.

49.

y

x

y

(45)

Chapter 7 Review

50. y

(46)

Sequences—Limit of a Sequence

1. no

2. no

3. yes

4. no

5. 0

6. 3

4

7. –2

8. no limit

9. 1.618