Other Course Material by AskDrCallahan

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1 Other Course Material by AskDrCallahan Algebra DVD Set Video teaching on DVD for Harold Jacobs s textbook Elementary Algebra. Geometry DVD Set Video teaching on DVD for Harold Jacobs s textbook Geometry: Seeing, Doing, Understanding, Third Edition. Algebra II with Trigonometry Video teaching on DVD for Raymond Barnett s College Algebra with Trigonometry. Can be used for Algebra II or Trig alone or as Precalculus. Calculus Video teaching on DVD for James Stewart s Calculus: Concepts and Contexts. Equivalent to Calculus at most universities. Discovering the Entrepreneur in You Audio course based on the years of coaching startup companies to find success. See website for more details. Website: products@askdrcallahan.com

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3 SOLUTIONS MANUAL FOR Harold R. Jacobs with Cassidy Cash AskDrCallahan, LLC

4 ISBN: ! 9 by Harold R. Jacobs. All rights reserved. Printed in the United States of America First printing 9

5 Introductory Comments The exercises for each lesson in Elementary Algebra are arranged in four sets. Those in Set I review ideas from preceding lessons, affording continual practice with material considered earlier in the course. Sets II and III enable the student to apply the concepts of the new lesson. Because both sets have the same content, you will probably prefer to work one set or the other. Answers to all of the Set II exercises are given at the back of the textbook. Those answers, together with those to Sets I, III, and IV, are also included in this book. Solutions to many of the exercises appear in brackets following the answers. These solutions vary in detail according to the complexity of the exercise. For those exercises that can be solved in a variety of ways, just one example solution is given. Most of the Set IV exercises are intended to challenge the better students, although some are simple enough that everyone can be successful with them. They should be considered optional and counted as extra credit. Detailed lesson plans for the course are available in the Teacher s Guide to Elementary Algebra. I am grateful to Dale Callahan and Cassidy Cash for their help in preparing and producing this solutions manual. Harold R. Jacobs

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7 ANSWERS TO EXERCISES Introduction (pp. ). a) Think of a number: 5 Double it: 6 8 Add six: 8 6 Divide by two: Subtract the number you first thought of: No. c) Think of a number: Double it: Add six: Divide by two: Subtract the number you first thought of: d) Yes.. a) Think of a number. Multiply it by four. c) Add eight. d) Divide by four. e) Add three. f) Subtract the number that you first thought of.. a). The result at the end is instead of. c) The result at the end is. d) The result at the end now depends on the number first thought of.. (One of many possible answers.) Think of a number. Triple it. Add twelve. Divide by three. Subtract the number that you first thought of. The result is four. Set I (page ).,776.., ,7. 6., Set II (pages ). a) + 7 or 7. x + 7. c) + y. d) x + y. e) + 8 or. f) + z. g) or 8. h) x or x + 6. i) + y + or y +. j) x + y +.. a) c) x + 5. d) 7. e) 9.. a) or. x + 6. c) x + y. d) x or 8 + x. e) x + + y + or x + y +. f) x + y + z.. a) and and c) and 5. a) 8 + y + or y y + or y +. c) x + + or x + 5. d) x + + or x +. e) or. 6. a). 9 + x. c) 9 + x + 6 or x + 5. d) x + 5. e) x + y. f) x + y + z. Set III (page ) 7. a) + or. + x.

8 c) y +. d) y + x. e) 7 + or 9. f) 7 + x. g) or. h) x + + or x + 5. i) 9 + y + or y +. j) x + y a) c) x + 6. d) 7. e). 9. a) 7. x +. c) x + y + 6. d) x + x + x. e) x + y + x + y.. a) and and c) and. a) + + y or + y. + + y or + y. c) x or x + 7. d) x or x +. e) or.. a) 9 miles. x + miles. c) y + miles. d) y + z miles. Set IV (page ) It would happen with any four numbers because: a b a + b x y x + y a + x b + y a + b + x + y = a + x + b + y Set I (page )., , Set II (pages ). a) 7 or. 6 x. c) x 6. d) or 8. e) x. f) x y. g) x. h) x.. a) 7 or 5. x. c) x. d) y x. e) 9 or. f) x y.. a).. c). d) The value of x gets larger. e). f). g) 5. h) The value of 5 x gets smaller.. a) 8.. c) 8. d). e) Each expression is x + y. 5. a). 7 x. c) 6. d) y. 6. a) 7, x pounds. 7, + y pounds. 7. a) cents. y x cents. c) x + cents.

9 d) 95 y cents. Set III (pages ) 8. a) 9 or 6. x 5. c) 5 x. d) 8 or. e) x. f) y x. g) 7 x. h) x a) 5. y 7. c). d) y. e) y 5. f) z y.. a) c) 8. d) The value of x gets larger. e) 7. f) 6. g). h) The value of 8 x gets smaller.. a).. c). d). e) Each expression is x + 7 y.. a) x feet. x + y feet. c) x y feet.. a) x dollars. + y dollars. c) z dollars.. a) x. + y. c) + y z. Set IV (page ) The clerk is giving a customer the change for an $8.7 purchase: $ $8.7 = $.5. The problem is being solved by addition. Set I (page ). 6,.. 7, , ,. Set II (pages ). a) for! and for!. c). a) 5! 6 or or. c) 5x. d) 5 + x. e) xy. f) x + y. g) xx. h) 8x. i) x 8. j) x or 9 + x. k)! 7! x or x. l) + y + or y +. m)! y! or y. n) + x + y. o) xy.. a) 6!.! 6. c) 5x. d)! 7. e) x! 7 or 7x. f) xy. g) h) x + x + x + x. i) + +!!! + (y of them). j) z + z +!!! + z (y of them).. a) 7! 8 or 56. x. c) xy.

10 d) xx. 5. a). 5x. 6. a) 7x. x. c),. d),x. e),8x. f) x. g),x. 7. a) 65. x. Set III (pages ) 8. a) c) 6x. c) 7xy. d) xxx.. a) 8 cents. 9x cents. c) 8y cents.. a) x. xx.. a),.,x. c),. d),y. e),. f),y. g),,y. Set IV (page ). Yes: 95 is correct. 9. a).. c) 7x. d) 7 + x. e) xy. f) x + y. g) yy. h) 5x. i) x 5. j) + x. k) x. l) y + 7. m) 6y. n) x + y +. o) xy.. a)!.!. c) x. d) 5!. e) x. f) xy. g) h) x + x + x + x + x. i) + +!!! + (y of them). j) x + x +!!! + x (x of them).. a) 7.. Set I (page ) Set II (pages ) 9. a) or. or 9. Yes. If we carry from each slanting column to the next going clockwise, we get 7.

11 c). d). e). f) x! or x. g). h) x. i). j) xy.. a) = and =. =. c) = 6 and =.. a) 5! = 5.! = 9. c)! =. d) (7.5)() = 7.5. e)! 7 = x. f) x! = 6. g) x =. h) y! = x.. a) 5. [9! 5.] 6. [9! 7.] c) 8. [9!.] d) The value of 9x gets larger. e). [.] f) 5. [.] g) 5. [.] h) The value of gets larger. i) 5. [.] j) 6. [.] k).5. [.] l) The value of gets smaller.. a). [.].. a) 7x dollars. 6. [.] c). 5. a) x inches or x feet. 5. [.] c). 6. a).6. [.]. Set III (pages ) 7. a) or. 8 or 6. c). d). e). f) x. g). h) x.

12 i). j) yx. 8. a).. c) 77. d) The value of 7x gets larger. e). f). g) 7. h) The value of gets larger. i). j).8. k).. l) The value of gets smaller. 9. a) 7. [.] 8. [.] c). d). e).. a) twenty-ones more twenty-ones 6 twenty-ones subtracted 5 7) sevens more sevens more sevens 5 sevens subtracted. a) minutes. [.] minutes.. a) xy.. c).. a) 5 square inches. [6! 5.] 6x square inches. c) 6 square inches. [.] d) square inches.. a).8 dollars. [.].9 dollars. [.] c) dollars. d) dollars. Set IV (page ) There were twelve loaves of bread altogether. In sharing them equally, the three travelers each got four loaves. This means that the third traveler got three loaves from the pilgrim who had seven loaves and one loaf from the pilgrim who had five loaves. Because = of the traveler s bread came from the first pilgrim, he should receive of the twelve coins: of coins = 9 coins. Set I (page ) ,..,,

13 Set II (pages ). a) x squared and x to the second power. An exponent.. a). 5. c) x. d). e) x.. a) c) x. d) x 8. e) x. f) x y.. a) c) x 6. d). e) x. f) x y. g) 8! 8! 8! 8! 8. h) x! x! x. i)!!!!!!! (x of them). j) y! y!!!!!! y (x of them). 5. a), a) c). d) 8. e). f) 9. g) Because all powers of are equal to. 7. a) 5. [! 56.],6. [!,.] c),87. [!!.] d) 9,65. [5! 5! 5!,5.] e) x. f) x. Set III (pages ) 8. a) x cubed and x to the third power. An exponent. 9. a) 6.. c) y. d) 5. e) y.. a). 5. c) x. d) x. e) 9 y. f) y x.. a) c) y 7. d). e) x. f) x y. g)!!!. h) x! x! x! x! x. i) 5! 5!!!!!! 5 (x of them). j) x! x!!!!!! x (y of them).. a) c) 9. d) 8. e). f) 7. g) Because all powers of are equal to.. a) The last digit of 6 is 6. 5 = 5 5 = 5 5 = =,5 5 6 = 5,65 c) The last two digits of 5 are 5. d) The last digit of 9 is.

14 Set IV (page ) This table shows Ollie s study times: Week Minutes! =!! = 5!!! = 6!!!! = According to Ollie s plan, he would study 9 minutes in the last week, but 9 = 5,88 minutes. Altogether there are 7!! 6 =,8 minutes in a week, so Ollie wouldn t be able to do it! Set I (page ). a) c) d) e) f). a) x. + x. c) x. d). e) x. f) x.. a) a. b + b + b + b + b. c) c. d) d! d! d! d. e) xe. f) f y. Set II (pages ). a) The sum of a number and zero is the number. The difference between a number and zero is the number. c) The product of a number and zero is zero. d) The product of a number and one is the number. e) The quotient of zero and a nonzero number is zero. f) The quotient of a number and zero is not defined. g) The quotient of a number and one is the number. 5. a) 5.. c). d). 6. a) = because times is.. c). 7. a). has a higher value than. 8. a) x.. c) x. d) x + cannot be simplified. e) x. f). g) is not defined. h) x. 9. a) x + y. x. c) y. d). e) x + y. f). g) x. h) y.. a).. c) Even. Set III (pages ). a).. c). d). e) is meaningless.

15 f). g) x. h).. a) The product! 5 would have two rows. The product! 5 would have one row. c) No;! 5 would have no rows.. a) because zero times zero is zero.. c).. a) x. y x. c) y. d). e) y + x. f). g) y. h) x. 5. a) 5 5 The answer is tens and ones. 6. a) x +. x. c). 7. a) If were equal to, then! would be equal to 7. No. If were equal to 7, then! 7 would be equal to 7. c) = because 7! =. d) No; because we could also say = a for any number a. 8. a) No, because no matter how many times zero is subtracted from, the result is always. The calculator endlessly subtracts zero from zero without getting any answer. [This is the way that electrically driven calculators work. An electronic calculator will give an error message.] Set IV (page ) If x is not equal to, then x =. If x is equal to, it would seem that =. Set I (page ). a) 5. Not possible. c) 6.. a) x +. 7 x. c) x a) 7x. days. c) 5, + x. Set II (pages ). a) Figure. Figure 5. c) Figure. d) Figure 6. e) Figure. f) Figure. g) Figure. 5. a) 5. [ + = 5.]. [ + + =.] c) 8. [! 6 = 8.] d) 9. [ + 6 = 9.] e) 9. [5 6 = 9.] f). [5! 6 =.] g) 57. [ + 7 = 57.] h) 57. [ + 7 = 57.] i) 7. [ 7 + = 7.] j) 7. [ + 7 = 7.] k) 8. [ + = + 6 = 8.] l). [ + = + 8 =.] m) 8. [ + = 6 + = 8.] n) 9. [ 6 + = 9.] o) 5. [ = 5.] p) 9. [! 8 9 = 88 9 = 9.] q) 5. [! 9 8 = 59 8 = 5.]

16 6. a) x + y. 5x. c). d) 8x. e) y y. f) +. g) x + xy. 7. a). [ +! = + =.] 6. [ +! = 6 + = 6.] c) 8. [ +! = + = 8.] d) 58. [ +! = + 6 = 58.] 8. a) 9. [! = + 7 = 9.] 9. [5! = 5 6 = 9.] c). [ +! 5 = +! 5 = + =.] d) 9. [ = = 9.] e) 8. [ + = 8 + = 8.] f) 8. [5! + 6 = 5! = = 8.] 9. a) 85 cents or $8.5. [7! 8 +! 95 = = 85.] 8x + 95y cents. Set III (page ). a) Figure. Figure 6. c) Figure. d) Figure. e) Figure 5. f) Figure 6. g) Figure.. a) 7. [ + = 7.]. [ 8 =.] c) 7. [ = 7.] d) 7. [5 8 = 7.] e). [ =.] f) 66. [ + 6 = 66.] g) 8. [! 6 = 8.] h) 8. [6! = 8.] i) 5. [6 + 5 = 5.] j) 5. [6 5 + = 5.] k) 7. [6 + 5 = 7.] l) 7. [6 + 5 = 7.] m). [ = =.] n).5. [ =.5 =.5.] o) 6. [6 = 6.] p) 7.! 8! = 96 = 7.] q) 7. 8!! = 96 = 7.]. a) 7x +. x y. c) x y. d) +. e) 5x. f) x + y 5. g).. a). [! + = + =.] 8. [! + = 8 + = 8.] c) 9. [5! 5 + = 5 + = 9.] d). [7! 7 + = + =.]. a). [5! 8 + = + =.]. [(7! = 7 6 =.] c) 9. [ +! = +! 6 = + 8 = 9.] d),. [ + = +, =,.] e) 6. [5 5 = 65 5 = 6.] f) 5. [6! + = 6 + = 5.] 5. a) 5 square centimeters. [ 7 = 9 = 5.] y x square centimeters. Set IV (page ). 7. [6 + = 7.].. [! 5 = 6; 6 8 = 5; 5 " = ; + 7 = ;! =.]. The calculator would do the operations in order from left to right.. Write down some of the intermediate steps. Set I (page ). a).. c). d) Not possible. e)..

17 f).. a) 7. [.] 6. [.] c) x. [.] d). e) x. [.] f). a) 5. [.]. Set II (pages ). a) Yes. [6 + = 8; + 7 = 8.] No. [6 = ; = 8.] c) Yes. [6 = ; + =.] d) No. [6 + = 8; 7 =.] e) Yes. [7! = 6;! 8 = 6.] f) Yes. [ + 8 = ; + 8 =.] g) No. [ + 8 = ; 8! = 5.] h) Yes. [ = 6; = 6.] 5. a) 6. [7! 9 = 6.]. [ =.] c) 5. [ = 5.] d). [6! + 5 = =.] e). [ +! 8 = + 6 =.] f) 8. [6! 8 = 8.] g). [5 =.] h). [! =.] i) 5. [5 ( ) = 5 = 5.] j) 9. [5! = 5 6 = 9.] k) 6. [ + = 6.] l). [ =.] m) 9. [! = 9.] n) 9. [ = 9.] o) 5. [5 5! = 5 = 5.] p) 8. [(5 5)! =! = 8.] q) 5. [(5 ) = 5 = 5.] 6. a) Figure. Figure. c) Figure. d) Figure 5. e) Figure. f) Figure 6. g) Figure. h) Figure a) (x 5). x! 6 + y or 6x + y. c) (y + 6)x. d) y. e). f) (x + )(x + 7). g). h) (x). i) ( x). j) (x + y )8 or 8(x + y ). 8. a). ( +! 5 = =.) 9. ( +! 5 = = 9.) c) 5. [ +! 5 = + 5 = 5.] d),585. [5 +! 5 5 =,5 + 5 =,585.] e). [! 8 =.] f) 9. [! 9 = 9.] g) 5. [7! 5 = 5.] h),585. [7! 55 =,585.] Set III (page ) 9. a) Yes. [ + = ; + 7 =.] Yes. [ = 9; + 5 = 9.] c) No. [8 = 7; 5 = 9.] d) No. [8 + = 9; 7 = 7.] e) Yes. [! = 8;! 8 = 8.] f) No. [ + 8 = 8;! = 8.] g) Yes. [( + 8 = 8; + 8 = 8.] h) Yes. [ = ; =.]. a) 8. [! 6 = 8.]. [ =.] c) 9. [8 + = 9.]

18 d) 85. [! 8 = 88 = 85.] e). [8 +! 5 = =.] f) 55. [! 5 = 55.] g). [ =.] h). [! =.] i). [ ( ) = 9 =.] j). [! = 8 =.] k) 6. [8 = 6.] l). [ =.] m) 6. [8! = 6.] n) 6. [ = 6.] o). [9 9! = 9 6 =.] p) 6. [(9 9)! =! = 6.] q) 96. [(9 8) = = 96.]. a) Figure. Figure. c) Figure. d) Figure 5. e) Figure. f) Figure 6. g) Figure. h) Figure 6.. a) (x + )y. y + x. c). d). e) (x + y). f) x + y. g) (x y)x. h) 9 (x). i) (9 x). j).. a). [ +! = + 8 =.]. [ +! = =.] c) 8. [ +! = + = 8.] d) 7. [5 +! 5 = = 7.] e). [8! =.] f). [! =.] g) 8. [6! 8 = 8.] h) 7. [! = 7.] Set IV (page ). All multiplication signs except for the first one which should be an addition sign: +!!! 5! 6! 7! 8! 9!..,68,8.. Use the same symbols of operation as before, but put parentheses around the + : ( + )!!! 5! 6! 7! 8! 9!.. 5,,. Set I (page ). a) 5a. b. c) a + b. d).. a) + x. 5x. c) minutes.. a) x pounds. pounds. c) z pounds. Set II (pages ). a) (6 + ) = (6) + (). (7 ) = (7) (). c) 5( + 8) = 5() + 5(8). d) 6(5 ) = 6(5) 6(). 5. a) x. 7(x). c) (x + ). d) 9(x + y). e) x + x. f) x + x + x + x + x. g) (x + 7) + (x + 7) + (x + 7) + (x + 7). 6. a) (x + 5) = (x + 5) + (x + 5) + (x + 5) = x + x + x = x + 5. (x + y) = (x + y) + (x + y) = x + x + y + y = x + y.

19 c) (x + ) = (x + ) + (x + ) + (x + ) + (x + ) = x + x + x + x = x a) 8x +. 5y. c) x + x. d) xy y. e) x + 8. f) y + xy. g) 7y 7x. h) x 6x. i) x +. j) x x. 8. a) 7 #! ! 7 = ( + )7 =! 7 +! 7 = = 96 c) #! d) 7! = (7 + ) = 7! +! = + 86 = a) (x + 5) and x +. x( + x) and x + x. c) (x + y + ) and x + y + 9. d) x(x + x + ) and x + x + x.. a) x + y. (x + y). c) x. d) y. e) x + y. Set III (page ). a) (5 + ) = (5) + (). 7( + ) = 7() + 7(). c) (8 5) = (8) (5). d) 5( ) = 5() 5().. a) 7x. (5x) or 5x. c) (x + 7). d) (x + y). e) x + x + x + x. f) 7x + 7x. g) (x + 8) + (x + 8) + (x + 8).. a) (x + 6) = (x + 6) + (x + 6) = x + x = x +. (x + y) = (x + y) + (x + y) + (x + y) + (x + y) = x + x + x + x + y + y + y + y = x + y. c) (x + ) = (x + ) + (x + ) + (x + ) = x + x + x = x a) x +. y 8. c) x + x. d) y y. e) x + 8. f) 6x + x. g) 5y. h) xy y. i) x + 7. j) x x. 5. a) 8 #! ! 8 = ( + )8 =! 8 +! 8 = = 76 c) #! d) 8! = (8 + ) = 8! +! = = a) (x + 7) and x +. y( + y) and y + y. c) (x + y + 6) and x + y +. d) x(y + y + ) and xy + xy + x.

20 7. a) x + y. (x + y) cents. c) x cents. d) y cents. e) x + y cents. Set IV (page ). a). [( + ) = =.] 6. [( + 6) = 6 = 6.] c) 9. [( + ) = 7 = 9.] d). [(9 + ) = =.] e). [ + = + =.] f) 6. [ + 6 = + 6 = 6.] g) 5. [ + = = 5.] h) 8. [9 + = 8 + = 8.]. (x + y) and x + y are sometimes equal and sometimes not equal. Chapter, Review Set I (pages ). a)! c) x + x. d) y! y! y! y! y! y.. a) w. x. c) 7 y. d) z 5.. a) Think of a number: Multiply by five: Add eight: Subtract three: Divide by five: Subtract the number you first thought of: The number. c) Steps and. d) Add 5.. a) Figure. Figure. c) Figure. 5. a) = 6 = 6 = 56 5 =, 6 =,96 The last digit of is a) a) Perimeter, ; area, 8. Perimeter, x + 6; area, x. c) Perimeter, y; area, y. 8. a). [ 7 =.] 8. [ = 8.] c) 65. [ + 6 = 65.] d) 5. [5 = 5.] 9. a) x + 5. [9 + x +.] 6 x y or 6 (x + y).. a) If = a, then! a =. But! a =. No.. a) 6x... a) 5x +. (x + ). c) x a) 7a +. b b. c) 5c a) x carbon catoms and 8x hydrogen atoms. x + 5x. c). d) x. Set II (pages ). a)!. 7. c) x + x + x + x + x. d) y! y! y! y.. Step. Think of a number. Step. Add. Step. Multiply by. Step. Add 8. Step 5. Divide by. Step 6. Subtract the number that you first thought of.

21 Chapter, Review. a) a 5. b. c) + c. d).. a) 5. [ =, =, = 8, = 6, 5 =.] It is impossible to express as a power of. [ =, =, =,...] c) 6. [ =, =, =,, =,, 5 =,, 6 =,,.]. a) x. x kilograms. c).. a). [ + 5! = + =.]. [ + 5! = =.] c) 6. [ + 5! = + 5 = 6.] d). [9! =.] e). [! =.] f) 6. [7! 8 = 6.] 5. a) x. x a) 9. [.]. c). 7. a) 7 + x. 7 y. c) + x. [75 + x 7.] 8. a) 6. [6! = 6.],6. [6 =,6.] c) 5. [9! 5 = 5.] d) 55. [ + 56 = 55.] 9. a) = 8 = 6 = 7 = 6 = 6 = 5 = 5 = 5 5 = 6 = 6.. a) (x 6) or (x 6). +. c) 5 x.. a) 8v w 8. c) xy + xz.. a) (x + ) = x +. (6 + x)x = 6x + x. c) y(x + ) = yx + y.

22 Set I (pages ). a). [ 9 =.] 8. [9 = 8.] c) 9. [ + 8 = 9.] d) 7. [ = 7.]. a).. c).. a) x. x. Set II (pages ). a) x y x 6 8 y 8 6 c) x y d) x 5 y e) x 6 8 y a) x 5 y x 5 y a) y = 6x. y = x + 7. c) y = x. d) y = x. e) y = x +. f) y = 6x. g) y = x +. h) y = x. i) y = x. j) y =. 7. a). [! =.] p = s. c). [ =.] d) a = s. 8. a) meters per second. [ = meters per second.] m = s. c) 77 meters. [! 7 = 77.] 9. a) The population is increasing. No. Set III (page ). a) x y 5 5 x y c) x 5 y 5 d) x 5 y e) x y 6. a) x y 6 x 5 y a) y = x +. y = 5x. c) y = (or y = x.) d) y = x 6. e) y = x. f) y = 5x. g) y = x +. h) y = x +. i) y = x. j) y = x.. a) The longer the senator spoke, the fewer the people who were listening to him. No.. a) 7.5 dollars. [ = 7.5 dollars per hour.] d = 7.5h. c) 75 dollars. [7.5! 5 = 75.] 5. a)..

23 c) n = d. Set IV (page ). Taking first and second differences of the temperatures, we get s () t,5 6,,5 8,8 6,5 (5,). 5,!C.,95 5,85 6,75 7,65 (8,55) (9) Set I (pages ). a) 5n. n 5. c) 7! 7! 7! 7. d) a). [.] x. [.] d) They lie on a straight line. 7. a) x y y = x a) x y y = 7 x. 9. a) x 6 8 y 8 6 c).. a) x 9. (x 9) or (x 9). c) a + b. d) (a +. Set II (pages ). O (, ); A (, ); B (6, 6); C (, 7); D (7, ); E (8, ); F (, 5).. a) x y 7 5. a) 6. a) y = x. c) Four. Set III (pages ). A (, ); B (5, ); C (8, ); D (, 5); E (9, 8); F (, 9); G (, 5); O (, ).

24 #& Chapter!" Lesson!. a) Set IV (page %%). a) y = x. Four. c) Chapter!" Lesson * Set I (page &+). a) x y a) c) d) Yes. No. Yes. No.. a) c) d) x + y. 5x 5. x x. y + y a) 6x. x. y = x. 5. a) x y 5 y = 5 x. 6. a) x y [ + = 6; = 6.] [5 + = 8; 9 7 =.] [ = ; 9 + =.] [5 = ; 9 = 8.] Set II (pages &+ &#). a) x y y = x. 5. a) x =. y = and y =. 7. a) x y a) c) d) e) f) g) h) i) j) Yes. Yes. No. Yes. Yes. Yes. No. No. Yes. Yes. 6 9

25 Chapter!" Lesson * 7. a) x y x y 5 6 c) x y 6 8 g) x y h) x y Set III (page &!) 8. a) x y 5 6 y = x a) x =. y = and y =. d) x y 5 7 e) x y 9 f) x y 6. a) c) d) e) f) g) h) i) j). Yes. Yes. No. Yes. Yes. No. Yes. No. Yes. No. a) x y x y #'

26 . c) x y 7 Set IV (page ). There is more than one value of y corresponding to x =.5, x =, x =, x =.5, and x =.5.. Remove the vertical line segments and either the right or left endpoints of the horizontal line segments. (Example: ) d) x.5.5 y e) x y 5 7 f) x y 5 g) x 6 8 y.5.5 Set I (page ). a) Figure. Figure. c) Figure. d) Figure.. a) y = 6 x. y = x +. c) y = x.. a) x minutes. y. Set II (pages ). a) x y 5 5 y is tripled. c) A direct variation. d) y = 5x. e) 5. [y = 5! 7 = 5.] 5. a) x 8 y c) y = x. h) x 5 y a) y =. y = a! =. c) (, ). d) The line goes through the origin.

27 7. a) Yes. No. c) Yes. d) No. e) No. 8. a) x y 6 x y 6 9 c) x y 8 e) for y = x; for y = x; for y = x. f) y = x. g) The steepness increases as the constant of variation increases. 9. a) 75 centimeters. [.5! 5 = 75.] centimeters. [ =.5!?] c) The bounciness increases as the constant of variation increases. d) No; in that case the ball would bounce higher and higher.. a) y = 5x. A steep line going through the origin. Set III (pages ). a) x y 6 9 y is doubled. c) A direct variation. d) y = x. e). [y =! =.]. a) x 6 8 y 7 6 or.5. c) y = x or y =.5x.. a) Yes; the speed of light is the constant of variation. No; after awhile the cat stops growing. c) No; too many cooks and there is no soup. d) d) Yes, if the bottles are uniform in size; the constant of variation is the refund per bottle.. a) Yes. No. c) No. d) Yes. e) No. 5. a) y = x. y = x. c) y = x. d) y =.5x. e) for line ; for line ; for line ;.5 for line. f) Line. g) The larger the constant of variation, the steeper the line. 6. a) 6 cubic centimeters. [y =! = 6.] 5 degrees Kelvin. [5 =!?] c) The greater the constant of variation, the greater the size of the balloon. 7. a) y =.x. A straight line passing through the origin. Set IV (page ). There is a discount if you buy more than cassettes.. It would be cheaper to buy. Set I (page ). a) x y. x +. [x + + x = x +.] c) y +. [x + y + x = y +.]. a) x. x. c) 8x.. a) + = = 5; 5 = 5. True = = 6; 6 = 6. True. c) = ,96 =,58; 7 =,. False.

28 Set II (pages ) Set III (pages ). a) x y 5 d). a) x y 5 7 d) x y x y 6 8 c) x y c) x y e) They are parallel. f) y = x + at (, ); y = x + at (, ); y = x + 7 at (, 7). g) At (, ). 5. a) x y 5 7 x y 7 c) x y 5 9 d) e) They are parallel. f) y = x + at (, ); y = x + at (, ); y = x + 5 at (, 5). g) At (, 8).. a) x y 5 6 x y c) x y 6 9 d) e) They all pass through the same point. f) At (, ). g) At (, ). e) They all pass through the same point. f) At (, ). g) At (, ). 6. a) x y a) x y c) It is a horizontal line. d) y = 5. c) It is a horizontal line. d) y =. 7. a) It is the difference between successive values of y. It is the value of y corresponding to x =. c) y = x + 8. d) y = 7x +. e) y = x + 6. f) y = 6x a) y = x + 6. [y = x + + x +.] Linear. c) No. 9. a) x y.5.5 centimeters. c).5 centimeters. d) centimeters. [y =.5! + = + =.]. a) It is the difference between successive values of y. It is the value of y corresponding to x =. c) y = 6x +. d) y = x +. e) y = 5x + 9. f) y = 9x a) y = x + 5. [y = x + x + 5.] Linear. c) No. 5. a) x 5 y y =.5x c) No. d) $.5. [y =.5! = =.5.]

29 Set IV (page ). 6x +. [One way to figure this out is to extend the table to the left: x 5 y 6 8 The number 6 is the difference between successive values of y. The number is the value of y corresponding to x =.].!C. [The ant stays home when its speed, x, is equal to. When x =, y =.] Set I (page ). a) x. (x + 5). c) x.. a) The axes. The origin. c) Coordinates.. a) Directly. A straight line passing through the origin. Set II (pages ). a) x 5 6 y 6. y is divided by. c) An inverse variation. d) y =. e).. [y = =..] 5. a) x 6 y c) y =. 6. a) y = a.. c) We cannot divide by. d) No. [Because there is no point for which x =.] 7. a) Yes. No. c) No. d) Yes. e) No. 8. a) x 5 y..8 x 5 y 6.5. c) x 5 y 5..5 d) e) for y = ; 6 for y = ; for y =. f) The larger the constant of variation, the farther the curve is from the origin. 9. a) y =.. c) Inversely.. a) s =. c) It was approximately 9.9 meters per second. [ " 9.9 meters per second.] Set III (pages ). a) x 5 y 5 If x is doubled, y is halved. c) An inverse variation. d) y =. e).5. [y = =.5.]

30 . a) x 6 6 y 8 8. c) y =.. a) No; if this were true, a baby would have a very long attention span. Yes; if someone swims twice as fast, the time required is cut in half. c) Yes; if the price is doubled, you can buy half as much. d) No; the more dogs, the faster the sled.. a) Yes. No. c) No. d) Yes. e) No. 5. a) x 5 y x 5 y c) x 5 y 6. d) e) 5 for y = ; 8 for y = ; for y =. f) The larger the constant of variation, the farther the curve is from the origin. 6. a) y =. 6. c) Inversely. 7. a) n =. 66. [n = = 66.] Set IV (page ). f =. [Making this little table helps: d.] f kilograms. [f = = 7.5.] Chapter, Review Set I (pages ).. a) x 5 y 6 5 x y c) x y 5. a) True. True. c) False. [The graphs of all linear functions do not intersect the origin.] d) True. e) False. [It is ; y = x.]. a) y = x 5. y = x. c) y =. d) y = x a) x 6 8 y 5 y is tripled. c) A direct variation. d) y =. e) 5. [y = = 5.]

31 Chapter, Review 6. a) x y 9. a) x 5 6 y. x y c) Inversely. d) It becomes very small. Set II (pages ). c) x y a) x y 6 9 c) A direct variation. d) 7.5 seconds. [y =!.5 = 7.5.] 8. a) x 8 6 y c) No. d) No; the graph does not pass through the origin.. a) x 5 y x y 8 8 c) x 5 y 6 8. a) True. False. [The graphs of direct variations intersect the origin.] c) True. d) True. e) False. [Not all functions whose graphs are curved lines are inverse variations.]. a) y = 8 x. y = x. c) y = x. d) y = x or y =.5x 5. a) x y No. c) A linear function. d) y = x +. e) 5. [y =! 5 + = 5.]

32 Chapter, Review 6. a) x y a) x 6 8 y x y 7 5 c) x y 5 7 c) Inversely. d) It is impossible to divide by. [y = = no number.] 7. a) y =. Directly. c) A straight line going through the origin. 8. a) x 6 8 y karats. c) d) A direct variation.