Linear Functions. Solutions Key. 103 Holt McDougal Algebra 1. Are you Ready? 41 CHECK IT OUT!


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1 CHAPTER Linear Functions Solutions Ke Are ou Read? 1. E. C. A. B G A C F E 8  B 8 D H = = 81. = 68 = 68 = g  = ()  = 8  = = () + 8 = = 1. v =. +.1m 1. =  1 =  1 = = 1 + = + = 18. 8p  1 = 8()  1 =  1 =. t  1 = (1)  1 = = . Possible answer: The amount of mone in our bank account equals $1 minus the amount spent.. miles $1. = mi/gal. 1 gallons pounds = $.7/lb grams. servings = 8 g/serving 1 pictures 6. = pictures/roll rolls a. = = = 9 The equation can be written in standard form, so the function is linear. =  9 (, ) = ()  9 = 9 (, 9) 1 = (1)  9 =  (1, ) = ()  9 = 1 (, 1) Plot the points and connect them with a straight line b. = 1 + = 1 The equation can be written in standard form, so the function is linear. = 1 (, ) 1 = 1 (1, 1) = 1 (, 1) 1 = 1 (1, 1) Plot the points and connect them with a straight line.  8 c. This is not linear, because appears in an eponent. 1 Identifing linear functions CHECK IT OUT! 1a. Yes; each domain value is paired with eactl one range value; es b. Yes; each domain value is paired with eactl one range value; es c. No; each domain value is not paired with eactl one range value; no, not a linear function.. Yes; a constant change of + in corresponds to a constant change of 1 in. 1 Holt McDougal Algebra 1
2 . f() = + 1 (, f()) Rental pament ($) f() = () + 1 = 1 (, 1) 1 f() = (1) + 1 = 1 (1, 1) f() = () + 1 = 16 (, 16) f() = () + 1 = 19 (, 19) f() = () + 1 = (, ) f() = () + 1 = (, ) 6 f() = (6) + 1 = 8 (6, 8) 7 f() = (7) + 1 = 1 (7, 1) 16 8 Rental Pament 6 Manicures The number of manicures must be a whole number, so the domain is {, 1,,,...}. The range is {$1, $1, $16, $19,...}. think and discuss 1. No; all the points of the function must form a line in order for it to be a linear function.. It is onl possible to do a whole number of manicures, so the points whose coordinates are not whole numbers have no meaning in this situation.. From its graph: All the points form a line. Determining Whether a Function Is Linear From its equation: It can be written in standard form, A + B = C, where A and B are not both. Eample: 6 + =  From a list of ordered pairs: A constant change in corresponds to a constant change in Yes; a constant change of  in corresponds to a constant change of  in. 8. No; a constant change of  in corresponds to different changes in = The equation can be written in standard form, so the function is linear. + =   =  =  =  =  (, )  =  () = (, ) 1 =  (1) = 1 (1, 1) =  () = 1 (, 1) Plot the points and connect them with a straight line = 8 The equation can be written in standard form, so the function is linear. = 8 = 8 = Plot the points and connect them with a straight line. = (, ) 1 = (1, ) = (, ) 1 = (1, )  eercises guided practice 1. No; it is not in the form A + B = C.. Yes; each domain value is paired with eactl one range value; es. Yes; each domain value is paired with eactl one range value; es. Yes; each domain value is paired with eactl one range value; es. Yes; a constant change of 1 in corresponds to a constant change of + in. 6. No; there is a constant change in, but there is not a corresponding constant change in. 11. This is not linear, because has an eponent other than 1. 1 Holt McDougal Algebra 1
3 1. = = = The equation can be written in standard form, so the function is linear.  = = =   = = (, )  = ( ) =  (, ) = () = (, ) = () = (, ) Plot the points and connect them with a straight line f() = 7 (, f()) Distance (mi) f() = 7() = (, ) 1 f() = 7(1) = 7 (1, 7) f() = 7() = 1 (, 1) f() = 7() = (, ) f() = 7() = (, ) 16 8 Train Travel Time (h) The number of hours does not need to be a whole number, so the domain is. The range is. 1. f() =. + 6 (, f()) Cost ($) f() =.() + 6 = 6. (, 6.) 1 f() =.(1) + 6 = 8. (1, 8.) f() =.() + 6 = 11. (, 11.) f() =.() + 6 = 1. (, 1.) f() =.() + 6 = 16. (, 16.) f() =.() + 6 = 18. (, 18.) (, 6.) Movie Rentals (, 18.) (, 16.) (, 1.) (, 11.) (1, 8.) 6 8 Movies rented practice and problem solving The number of movies rented must be a whole number, so the domain is {, 1,,,...}. The range is {$6., $8., $11., $1.,...}. 1. Yes; each domain value is paired with eactl one range value; no 16. Yes; each domain value is paired with eactl one range value; es 17. Yes; each domain value is paired with eactl one range value; no 18. No; a constant change of + in corresponds to different changes in. 19. Yes; a constant change of +1 in corresponds to a constant change of +1 in.. Yes; a constant change of  in corresponds to a constant change of  in. 1. = + = The equation can be written in standard form, so the function is linear. = (, ) 1 = (1, ) = (, ) 1 = (1, ) Plot the points and connect them with a straight line.  1 Holt McDougal Algebra 1
4 .  =  + = The equation can be written in standard form, so the function is linear.  + = + + = = = 1 = 1 (, )  = 1 ( ) = 1 (, 1) = 1 () = (, ) = 1 () = 1 (, 1) Plot the points and connect them with a straight line. . This is not linear, because appears in the denominator of a fraction.. + = = + = The equation can be written in standard form, so the function is linear. = = = 1 = 1 (, ) 1 = 1 (1, 1) = 1 (, 1) 1 = 1 (1, 1) Plot the points and connect them with a straight line Gas left (gal) 1 f() = (, f()) f() =  1 () + 1 = 1 (, 1) f() =  1 () + 1 = 1 (, 1) f() =  1 () + 1 = 1 (, 1) 7 f() =  1 (7) + 1 = 1 (7, 1) 1 f() = (1) + 1 = 11 (1, 11) Ton s Drive 6 8 Distance driven (mi) The number of miles does not need to be a whole number. The maimum distance Ton can travel on 1 gallons is 1 gal mi = 7 mi, so the domain 1 gal is 7. The range is No; each domain value is not paired with eactl one range value. This is not a linear function. 7. Yes; each domain value is paired with eactl one range value; es; a constant change of + in corresponds to a constant change of  in. 8. Yes; each domain value is paired with eactl one range value; no; a constant change of +. in does not correspond to a constant change in. 9. Yes; each domain value is paired with eactl one range value; es; a constant change of + in corresponds to a constant change of + in = 16 The equation can be written in standard form, so the function is linear. A = ; B = 8; C = = =  + = The equation can be written in standard form, so the function is linear. A = ; B = 1; C = 16 Holt McDougal Algebra 1
5 . = = =  The equation can be written in standard form, so the function is linear. A = ; B =  1 ; C = . This is not linear, because appears in the denominator of a fraction. +. =  ( + ) = (  ) () + () = () + () + 1 = = =  The equation can be written in standard form, so the function is linear. A = ; B = ; C = . = 7 + = 7 The equation can be written in standard form, so it is linear, but it is not a function because there is more than one value of for. A = 1; B = ; C = 7 6. This is not linear, because and are multiplied together = = = 1 The equation can be written in standard form, so the function is linear. A = ; B = 1; C = 1 8. = = + = The equation can be written in standard form, so the function is linear. A = 1; B = 1; C = 9. = =  The equation can be written in standard form, so the function is linear. A = ; B = ; C = . = 6 + = 6 The equation can be written in standard form, so the function is linear. A = ; B = ; C = This is not a linear equation because appears in a radical sign.. = + 7 (, )  = () + 7 =  (, )  = () + 7 = 1 (, 1) 1 = (1) + 7 = (1, )  . = + (, )  =  + = (, )  = + = (, ) = + = 7 (, 7) 1. = 8  (, )  = 8  () = 1 (, 1) = 8  () = 8 (, 8) = 8  () = 6 (, 6) = (, ) 1 = (1) =  (1, )  = () = (, ) 1 = (1) = (1, ) 17 Holt McDougal Algebra 1
6 6.  = = =  =  (, ) = ()  =  (, ) = ()  = (, ) = ()  = (, ) = + + = + = + (, )  = () + = (, ) = + = (, ) = + = 6 (, 6) = = =  (, )  =   = 9 (, 9) =  =  (, ) =  = 1 (, 1) = 1a. f() = 8 A =.; B = 1; C = b. f() = 8 (, f()) Pa ($) f() = 8() = (, ) f() = 8() = 16 (, 16) f() = 8() = (, ) 6 f() = 8(6) = 8 (6, 8) 8 f() = 8(8) = 6 (8, 6) 1 Moll s Earnings The number of hours does not need to be a whole number, so the domain is. The range is = =  + =  + (, ) =  + () =  (, ) 1 =  + (1) = 1 (1, 1) =  + () = 1 (, 1) Time worked (h). Possible answer: = The table gives some ordered pairs (, ) that satisf the equation =  1. The graph is a representation of all ordered pairs (, ) that satisf = Possible answer: The value in cents of dimes is = 1. Since ou can have onl a whole number of dimes, the domain and range are restricted to whole numbers. a. Each constant change in time (+ minutes) corresponds with a constant change in calories (+7 calories). 18 Holt McDougal Algebra 1
7 b. Juan s Workout Calories burned Time (min) c. The graph forms a line.. No; each constant change in time (+1 s) is not accompanied b a constant change in height. 6. Yes; the equation can be written in standard form with A = 1, B =, and C = 9. No; all solutions are ordered pairs with value 9. The value 9 corresponds to more than one value. test prep 7. C; C is not linear, because appears in the denominator of a fraction. 8. G; Each second, sound will move 1 meters. So the distance covered is 1 times the number of seconds. 9. Possible answer: + = 7; it is a linear equation because it can be written in standard form with A =, B =, and C = 7. A table of values also shows it is a linear function: Each constant change in (+1) is accompanied b a constant change in (1.). The graph shows a linear function.  Both the graph and the table show solutions to the equation. challenge and etend 6. = ; = ; the first describes a linear function, but the second does not. 61. Perimeter of a Square 6.  Side Length Perimeter linear Volume of a Cube Side Length Volume not linear 6. Using Intercepts Check it out! Area of a Square Side Length Area not linear 1a. The intercept is. The intercept is . b.  + =  + () =  + =  =   =  = 1 The intercept is 1. c. + = 16 + () = 16 + = 16 = 16 = 16 = The intercept is.  + = () + = + = = = = 6 The intercept is 6. + = 16 () + = 16 + = 16 = 16 = 16 = 8 The intercept is 8. a. + = = 6  = 6  = = Notebooks School Store Purchases 1 1 Pens The intercept is. The intercept is. 19 Holt McDougal Algebra 1
8 b. intercept: pens that can be purchased if no notebooks are purchased. intercept: notebooks that can be purchased if no pens are purchased. a. intercept:  + = () = 1  = = 1  =  b. = 1  () = ( 1  ) = = 6 intercept:  + = () = 6  = 61() = 1(6) = 66 think and discuss 1. (, ) and (, )..18; Find the intercept b letting equal and solving for. Graphing A + B = C Using Intercepts. Find the intercept b letting equal and solving for. intercept:  + = 1 () + = 1 = 1 = 1 =  intercept:  + = 6 () + = 6 = 6 = 6 = . Graph the line b plotting the points containing the intercepts and then connecting the points with a straight line..  =  () =  = = = = The intercept is = = = () = = 6  = = 6  = The intercept is.  = ()  =  =  =   =  = 1 The intercept is = 6 ()  = 6  = 6  = = 6  = The intercept is = = = = = = () = 16 8() + = = 16 + = 16 8 = 16 = = 16 8 = The intercept is. = 16 = 8 The intercept is 8. 8a. 1 f() = Refrigeration Tank Temperature Temperature ( C) Time (h) 6 EXERCISES guided practice 1. intercept. The intercept is . The intercept is 1.. The intercept is. The intercept is .. The intercept is . The intercept is . The intercept is. The intercept is . b. intercept: time when temperature is C intercept: initial temperature 11 Holt McDougal Algebra 1
9 9. intercept:  =  () = = = = = = intercept:  + =  + =  =   =  = intercept: 11 = 11 () = 1 = ( 1 ) = () = intercept:  + = 1 () + = 1 = 1 = 1 =  intercept:  = ()  =  =   =  =  intercept:  + = () + = = intercept: 11 = 1 ()  1 =  1 =  (  1 ) = () = 8 intercept:  + = () = 1  = = 1  = practice and problem solving 1. The intercept is 1. The intercept is. 1. The intercept is . The intercept is The intercept is . The intercept is = () = = 1 6 = = 1 6 = The intercept is = = = = = () = = 8  = = 8  =  The intercept is = = 8  = 8  = 8  () = 8  = 8 = 8 The intercept is = 8 + = 8 = 8 = 8 = The intercept is..  = 1  = 1  = = 1  = The intercept is. 6 + = 1 6() + = 1 + = 1 = 1 = 1 = The intercept is.  + = 8 () + = 8 + = 8 = 8 = The intercept is.  = 8  = 8  = = 8  = The intercept is. + = 8 () + = 8 + = 8 = 8 The intercept is 8.  = 1  () = 1  = 1 = 1 The intercept is Holt McDougal Algebra 1
10 1. + = = = = 18 = = 18 = 1 8 The intercept is = 18() + = 1 + = 1 = 1 The intercept is 1. a f() = Population Bass Population 8 1 Time (r) The intercept is 1. The intercept is. b. intercept: time when bass population is intercept: number of bass originall in the lake a. 1 f() = Distance to finish line (km) 1 K Race 1 Time (min) The intercept is. The intercept is. b. intercept: total time to run the race (when the distance to the finish line is ) intercept: total length of the race (when time is ). intercept:  6 = 16() = 1 = 1 = 1 = intercept:  6 = 1 ()  6 = 16 = = 16 = . intercept: + = 18 + () = 18 = 18 = 18 = intercept: 1  = 1  () = 1 = ( 1 ) = () = intercept:  = 1  = 1  = 11() = 1(1) = intercept: + = 1 + () = 1 = 1 = 1 =  intercept: + = 18 () + = 18 = 18 = 18 = 6 intercept: = ()  =  =   =  = 1 intercept:  = 1  = 1 = 1 intercept: + = 1 () + = 1 = 1 = 1 = 11 Holt McDougal Algebra 1
11 9. intercept:  = 1  () = 1 = 1  a. = + b. = = intercept:  = 1  = 1  = = 1  = = () + = + = = The intercept is. c. The original height of the bamboo plant. 1a. The intercept is approimatel 7.. The intercept is approimatel 6. b. intercept: the number of ears after 18 when there will be no acres of tropical forest intercept: million acres of tropical forest in 18 a. m 6 8 b = 1  m Balance ($) 8 Account Balance 6 Time (mo) The number of months must be a whole number so the domain is (, 1,,,...}. The range is {$1, $8, $, $,...}. b. = = 1 + = 1 + = 1 + = 1 () + = 1 = 1 = + = 1 1 = 1 = 1 The intercept is 1. The intercept is 1. intercept: number of months from that time until the account has $ intercept: balance when bank emploee noticed the account c. After 1 months or 8 ears and 7 months. a. = 6 = = = 6: intercept: 6, no intercept = 1: intercept: 1, no intercept = : intercept:, no intercept b. = =  8 = 7 = : no intercept, intercept:  = : no intercept, intercept: = 7: no intercept, intercept: 7 c. Horizontal: For = c, the intercept is c and there is no intercept. Vertical: For = k, the intercept is k, and there is no intercept =   =  =   =  =  The intercept is = ()  =  =  = 1() = 1() =  The intercept is . Graph D has an intercept of  and a intercept of .. = = + = + = + = () + = = + = = = The intercept is. = The intercept is. Graph A has an intercept of and a intercept of = 8 () + = 8 + = 8 = 8 = 8 = The intercept is. + = 8 + () = 8 + = 8 = 8 = 8 = The intercept is. Graph A has an intercept of and a intercept of = 8  () = 8  = 8 = 8 = 8 = The intercept is.  = 8 ()  = 8  = 8  = = 8  =  The intercept is . Graph B has an intercept of and a intercept of . 8a. The intercept is. The intercept is 1.7. b. intercept: time remaining when Kristn started her workout intercept: total distance Kristn covered 9. Possible answer: Jen wants to save $6. Each week she will earn $1. The function shows how much mone Jen has left to save each week. 11 Holt McDougal Algebra 1
12 test prep. D; Notice that (9) = 18 =  18 = 9() So, (9, ) is on  = 918 and therefore is the intercept. 1. F; The intercept is  so Jamie owed her uncle $. The intercept is so Jamie was paing her uncle for weeks = 66 6() + = 66 + = 66 = 66 = 66 = 1 The intercept is 1. challenge and etend. intercept: = () = 1 1 = 1 ( 1 ) = (1) = . intercept:. . =.7. .() =.7. =.7.. =.7. = intercept: = 1 1 () + 1 = 1 1 = 1 ( 1 ) = (1) = intercept:. . =.7.() . =.7 . = =.7 . = .7. = = 6 intercept: = = 68 = 68 ( ) =  (6) = A + B = C A + B() = C A + = C A = C A = C A A = C A The intercept is C A =,98() =,9  =,9 =,9 =,9 = 9 The intercept is 9. intercept: = 68 () + = 6 = 6 A + B = C A() +B = C + B = C B = C B = C B B = C B The intercept is C B.  8 =,9 ()  8 =,98 =,98 =, =,98 =  The interecpt is . Possible answer: scale on the ais should include numbers from to a number a little greater than 9; scale on ais should include numbers from a little less than  to Holt McDougal Algebra 1
13  rate of Change and slope. Check it out! 1. dependent: balance independent: da da 1 to da 6 change in balance = 8  = 6 change in da 61 =  da 6 to da 16 change in balance = 18 change in da 166 = 7 1 = 7. da 16 to da change in balance = 11 change in da  16 = 6 = da to da change in balance = 171 change in da  =  8 = .7 The balance decreases at the greatest rate from da 1 to da 6.. Bank Balance Balance ($) 8 6 $/da $7./da $/da Positive: Negative: Zero: Undefined: Slope 1 $.8/da Da. slope =  =  a. rise run = 8 The slope is undefined. b. rise run = = The slope is. a. The slope is undefined. b. The slope is positive. think and discuss 1. 6 units; units; 6. decreased. Possible answer:, because it is less steep. eercises guided practice 1. constant. dependent: volume independent: time hour to hour 1 change in volume = 91 change in time 1  =  1 =  hour 1 to hour change in volume change in time = =  =  hour to hour 6 change in volume change in time = =  =  hour 6 to hour 7 change in volume change in time = = 1 = The volume decreased at the greatest rate from hour to hour 1.. Heart Rate 18 beats/min min 1 beats/min 1 min Heart rate (beats/min) beats/min min 6 8 Time (min)  beats/min min 11 Holt McDougal Algebra 1
14 . slope = 6 = 1. slope =  =  6. rise run = 6 = 7. rise run = 7 The slope is. The slope is undefined. 8. The slope is negative. 9. The slope is undefined. 1. The slope is zero. 11. The slope is positive. practice and problem solving 1. dependent: length independent: age month to month 9 change in length = = change in age 96 =.7 month 9 to month 18 change in length = =.1 change in age =. month 18 to month 6 change in length = change in age 618 =.9 8 =. month 6 to month change in length = change in age  6 =. 7 =. The bab increased in length at the greatest rate from month to month Elevator Movement 1. slope = = 1 m m s s 6 1. slope = 6 6 = 1 Distance (m)  m s 1 Time (s) m 7 s 16. rise run = The slope is undefined. 17. rise run = = The slope is. 18. The slope is positive. 19. The slope is positive.. Let l represent the length. slope = rise run.7 = 1 l.7 1 = 1 l.7l = 1.7l.7 = 1.7 l 1.7 The horizontal run that corresponds to a vertical change of 1 unit is Possible answer: slope is the ratio of change in to change in, and, for a line, it is alwas constant. a. slope =  = 1 b. This maimum heart rate decreases b 1 beat per minute ever ear.. slope = rise 8 run = 9 = a. 9 ft 16 ft 1, or.9 b. slope = rise run = Possible answer: The slope of a horizontal line will alwas be because the coordinates of an two points will be the same. Therefore, the numerator in the slope formula will alwas be. The slope of a vertical line will alwas be undefined because the coordinates of an two points will be the same. Therefore, the denominator in the slope formula will alwas be. Since ou cannot divide b, the slope will alwas be undefined. 6a. Road Trip Distance (mi) mi h mi h mi h mi h mi h 1 Time (h) b. The slope is greatest between hour and hour. Therefore, the rate of change is greatest between hour and hour. Therefore, the car s average speed was the greatest during the th hour. 7a. Possible answer: (16, ) b. Possible answer: (6, 6) c. Possible answer: change in files = 6  change in time 616 = 1 = 8a. Walk toward or awa from the motion detector at a constant rate. A line has constant slope, and in this case slope represents distance/time, or rate. So keeping the rate constant will result in a line. b. For a positive slope, walk awa from the detector. For a negative slope, walk towards the detector. c. Stand stillas time passes, our distance from the detector does not change. This graph is a horizontal line. test prep 9. C; The slope of line D is undefined so D is incorrect. Line C is the steepest so the absolute value of its slope is the greatest.. D; Since line D is a vertical line, it has a run of. 116 Holt McDougal Algebra 1
15 1. G; The slope of F is zero so F is incorrect. The slope of H is negative so H is incorrect. Choosing points (, ) and (1, ) on G give a rise of and a run of 1. So the slope of G is. challenge and etend. The slope of the hill is constant. Let r represent the rise of Jade s stride. slope of hill = Tara s rise Tara s run = 8 = 1 slope of hill = Jade s rise Jade s run 1 = r 6 6 = r 6 = r 9 = r Jade s rise is 9 inches. a. Electricit Costs Cost ($) kwh 1 kwh 1 kwh 1 kwh Energ used (kwh). kwh b. dependent: cost independent: energ kwh to kwh change in cost change in energ =   = = kwh to kwh change in cost change in energ = = 8 =.1 kwh to 6 kwh change in cost = 91 change in energ 6  = 8 =.1 6 kwh to 1 kwh change in cost = 119 change in energ 16 = 6 =.1 1 kwh to kwh change in cost = 111 change in energ  1 = 1 =. The rates of change for kwh to kwh, kwh to 6 kwh, and 6 kwh to 1 kwh are equivalent. c. The cost in dollars per kilowatt hour. d. Up to kwh costs $.. Between and 1 kwh costs $.1 per kwh. Between 1 and kwh costs $. per kwh.  the slope formula check it out! 1a. m = () = 7  () = 9 = c. m = = = 11 = b. m = = () = 6 =  d. Let (, ) be ( 1, 1 and (, ) be (,. m = = =  =  b. m = (7) = 6  = 1 = a. m = = = = 1 c. Let (, 1) be ( 1, 1 and (, ) be (,. m = = = =. m = = = 1 = 1 A slope of 1 means the height of the plant is increasing at a rate of 1 cm ever das.. Find the intercept. + = 1 + () = 1 = 1 = 1 = 6 m = = = Find the intercept. + = 1 () + = 1 = 1 = 1 = = Holt McDougal Algebra 1
16 think and discuss 1. values; values. vertical line. From a graph: Begin at an point on the line. Count rise and run to another point on the line. Slope is the ratio of rise to run. eercises guided practice 1. m = = = = 1. m = () = 9  (1) = 8 =  1 Finding Slope From a table: Choose an two points from the table and substitute their coordinates into the slope formula.. m = = =  = . m = From an equation: Find the  and  intercepts. Substitute the points containing the intercepts into the slope formula. =  (1)  () = 6 = 1. Let (, ) be ( 1, 1 and (, ) be (,. m = =   = = 1 6. m = = = 8 8 = 1 A slope of 1 means the mone earned is increasing at a rate of $1/h. 7. m = = = 6 = 1 A slope of 1 means for each jar of peanut butter, peanuts are needed. 8. Find the intercept: 8 + = () = 96 8 = = 96 8 = 1 m = = = 81 =  Find the intercept: 8 + = 96 8() + = 96 = 96 = 96 = 8 9. = = 9 Find the intercept: Find the intercept: + 9 = = 9 + 9() = 9 () + 9 = 9 = 9 = 9 = = 9 9 = 18 = 1 m = = = 1 = = = 16 Find the intercept: Find the intercept: 9 + = = () = 169() + = 169 = = = =  16 = 16 9 = m = =   (  16 = 9 ) 16 = 9 9 practice and problem solving 11. m = m =  1 = = =   () (9) 1 = 1 =  = 1. m = = =  The slope is undefined. 1. m = =  (1)  =  =  1. Let (, ) be ( 1, 1 and (, 9) be (,. m = = = Holt McDougal Algebra 1
17 16. m = () =  () = = 9 A slope of means the temperature in Celsius is 9 increasing at a rate of C for each 9 F. 17. m = = =.  =  9 A slope of  9 means the boiling point is decreasing at a rate of 9 F for each ft above sea level. 18. Find the intercept: = () = 91 7 = = 91 7 = 1 m = = = = = 1 Find the intercept: 1 + = () = 1 1 = = 1 1 = 1 m = = = = 9 + Find the intercept: 9 + = () = 7 9 = = 7 9 = 7 9 m = = Find the intercept: = 91 7() + 1 = 91 1 = = 91 1 = 71 = Find the intercept: 1 + = 1 1() + = 1 = 1 = 1 = 61 = 61 = Find the intercept: 9 + = 7 9() + = 7 = 7 = 7 =  7 = = 7 1. Student B is incorrect. Student B did not use the same coordinate pair order in the denominator as in the numerator. a. The rate of change for each interval is chirps/min 1 F. b. es; a. The distance of Car 1 is increasing at a faster rate than the distance of Car. So Car 1 is going faster. Since Car 1 traveled mi more in 1 h than Car, Car 1 is traveling mi/h faster than Car. b. The speed and the slope are both equal to the distance divided b time. c. Since Car 1 is traveling mi/h faster, the distance between the cars is changing at a rate of mi/h.. Possible answer: Given the points ( 1, 1 and (,, ou could substitute into the slope formula or graph the two points, connect with a line, and count the rise and the run. a. =  b. Maimum heart rate (beats/min) AgeBased Maimum Heart Rate Age (r) A slope of 1 means for each additional ear, the maimum heart rate decreases 1 beat/min. test prep 6. D; B finding the intercepts, ou obtain the points (, ) and (, ). B substituting into the slope formula ou obtain a slope of G; The slope of the line connecting (6, ) and (, ) is  1 so a line with slope of  1 could pass through these points. 8. 1, or. 9. m =  1 m = = b  =  a   (1) = 6 = 1. m = =   =  =  = b a =  b a 1. m = = =  = Holt McDougal Algebra 1
18 . m = = = (  ) = 6 + = = 1. m = =  () = + (1) = ( + )  = =. m = = = 1 = 6. Let (, ) represent the other point. m = = = (  )  1 = =  Since an point will do, let =. = ()  = 6  = So one possible point is (, ). 7. m = =   () =  = 1 m = = =  (1) = (  )  = =. m = =  ()  (1) 1 7 = = 7( + ) + 1 = = = = =  direct variation Check it out! 1a. = + 1 = + 1 = + This equation does not represent a direct variation because it cannot be written in the form = k. b. =   = = =  This equation represents a direct variation because it can be written in the form = k. The constant of variation is . c. + =   =  This equation represents a direct variation because it can be written in the form = k. The constant of variation is . a. No; possible answer: the value of is not the same for each ordered pair. b. Yes; possible answer: the value of is the same for each ordered pair. c. No; possible answer: the value of is not the same for each ordered pair.... = 1. = = 9. = = (, ) = () = (, ) 1 = (1) = (1, ) = () = 8 (, 8) Graph the points and connect. Perimeter 8 6 Perimeter of a Square 1 Side length think and discuss 1. It can written in the standard form k  = with A = k, B = 1, and C =. 1 Holt McDougal Algebra 1
19 . Possible answer: For an value of k, (, ) is a solution of = k.. Recognizing a Direct Variation From an Equation: The equation can be written in the form = k for some nonzero value of k. From Ordered Pairs: An equation describing the ordered pairs can be written in the form = k. Also, the ratio is constant for each ordered pair. From a Graph: The graph is a line through (, ). eercises guided practice 1. direct variation. This equation does not represent a direct variation because it cannot be written in the form = k.. = 8 = 8 =  This equation represents a direct variation because it can be written in the form = k. The constant of variation is =   =  This equation represents a direct variation because it can be written in the form = k. The constant of variation is 1.. No; possible answer: the value of is not the same for each ordered pair. 6. Yes; possible answer: the value of is the same for each ordered pair = = 1 = = 7 = 9. = 7 = 7 = 7() = 1 = 7(1) = 7 7 = 7() = 1 1 Graph the points and connect. Cameron s Wages practice and problem solving 1. This equation represents a direct variation because it can be written in the form = k. The constant of variation is = = = 1 This equation represents a direct variation because it can be written in the form = k. The constant of variation is = = + 1 = = = This equation does not represent a direct variation because it cannot be written in the form = k. 1. Yes; possible answer: the value of is the same for each ordered pair. 1. Yes; possible answer: the value of is the same for each ordered pair = 6  = 1 = =. 16. = 1 1 = 1 = 1 6 =. (, ) =.() = (, ) 1 =.(1) =. (1,.) =.() =. (,.) Graph the points and connect. Cost ($) 8 6 Cost of Gasoline Amount earned ($) Time worked (h) 6 8 Amount (gal) 18. Yes; it can be written as = No; it cannot be written in the form = k. 11 Holt McDougal Algebra 1
20 . = k 1 = k() = k The equation is =. = (, ) = () = (, ) 1 = (1) = (1, ) = () = 1 (, 1) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is = k 9 = k()  = k The equation is = . =  (, )  = () = 6 (, 6) 1 = (1) = (1, ) = () = (, ) Graph the points and connect. The value of k is , and the graph shows that the slope of the line is = k = k(8) 1 = k 1 The equation is = 1. = 1 (, ) = 1 () = (, ) = 1 () = 1 (, 1) 8 = 1 (8) = (8, ) Graph the points and connect. The value of k is 1, and the 1  graph shows that the slope of the line is 1.. = k 6 = k(1.) = k The equation is =. = (, ) = () = (, ) 1 = (1) = (1, ) = () = 8 (, 8) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is = k 1 = k(7) = k The equation is =. = (, ) 1 = (1) =  (1, ) = () = (, ) 1 = (1) = (1, ) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is = k = k(1) = k The equation is =. = (, ) 1 = (1) =  (1, ) = () = (, ) 1 = (1) = (1, ) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is Holt McDougal Algebra 1
21 6. = k 16 = k() 8 = k The equation is = 8. = 8 (, )  = 8() = 16 (, 16) 1 = 8(1) = 8 (1, 8) = 8() = (, ) Graph the points and connect. The value of k is 8, and the graph shows that the slope of the line is = k 1 = k ( 1 7 ) 7 = k The equation is = 7. = 7 (, ) = 7() = (, ) 1 = 7(1) = 7 (1, 7) = 7() = 1 (, 1) Graph the points and connect. The value of k is 7, and the 6 graph shows that the slope of the line is = k 9 = k()  9 = k 1 The equation is = =  (, ) 9  =  ( ) = 9 (, 9) 9 =  () = (, ) 9 =  () = 9 (, 9) Graph the points and 9 connect. The value of k is , and the graph shows 9 that the slope of 8 the line is = k  = k(9)  9 = k The equation is =  9. =  (, ) 9 =  () = (, ) 9 9 =  (9) =  (9, ) 9 18 =  (18) =  (18, ) 9 Graph the points and connect. The value of k is , and 9 the graph shows that the slope of the line is = k 6 = k() = k The equation is =. = (, )  = ( ) =  (, ) = () = (, ) = () = (, ) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is Holt McDougal Algebra 1
22 1. = k = k() = k The equation is =. = (, )  = ( ) =  (, ) = () = (, ) = () = (, ) Graph the points and connect. The value of k is, and the graph shows that the slope of the line is = k 1 = k() 1 = k  The equation is = 1. = 1 (, ) = 1 () = (, ) = 1 () = 1 (, 1) 1 = 1 (1) = (1, ) Graph the points and connect. The value of k is 1, and the = k 6 = k(1) 6 = k The equation is = 6. graph shows that the slope of the line is 1. = 6 (, ) 1 = 6(1) = 6 (1, 6) = 6() = (, ) 1 = 6(1) = 6 (1, 6) Graph the points and connect. The value of k is 6, and the graph shows that the slope of the line is = k 1 = k(1)  1 = k The equation is = =  (, )  =  1 ( ) = 1 (, 1) =  1 () = (, ) =  1 () = 1 (, 1) Graph the points and connect. The value of k is  1, and the = k = k(7) 7 = k  The equation is = 7. graph shows that the slope of the line is  1. = (, ) 7 = () = (, ) 7 7 = (7) = (7, ) 7 1 = (1) = (1, ) 7 Graph the points and connect. The value of k is, and the graph shows that the slope of the line is Let w represent its weight on Earth = w 1 91w = 118,88 w 9 The Mars rover weighed about 9 lb. on Earth. 7a. = 1 1 Holt McDougal Algebra 1
23 b. = 1 (, ) = 1() = (, ) 1 = 1(1) = 1 (1, 1) = 1() = (, ) Graph the points and connect. Washing Machine Efficienc No; possible answer: Mischa cannot wash 8 (6, 9) a fraction of a load of (, 7) laundr, so onl points 6 (, 6) whose coord. is a (, ) whole number make (, ) sense in this situation. (1, 1) (, ) 6 8 Loads of laundr Water saved (gal) c. loads 1 week 1 gal 1 load weeks = 16 gal 1 ear 8. Possible answer: Yes; since the ratio is the same for all ordered pairs, must correspond to. 9. Possible answer: The ratio is the same for all ordered pairs in a direct variation, so ou can write a proportion using an two ordered pairs. a. = b. It is written in the form = k, where k =. This value represents the speed at which Rhea is walking. test prep 1. C; = + 1 cannot be written in the form = k.. F; In F, the value of is the same for each ordered pair, so it is a direct variation.. B; Since 1 =. = 6., B is correct... Let h represent the number of hours. 18 = h 7 18h = 81 h =. b. Gas Mileage Distance (mi) 8 6 Hbrid SUV 6 Gas used (gal) c. = 1, = 1, = 7 = No; the lines begin at (, ) and then move awa from each other. = 6 1, = 6 1, 6 = 6 6 = 6. a + b = c a a b = a + c b = + c a b a b =  + c b b For the equation to be a direct variation, it must be able to be written in the form = k. So c = if it is a direct variation. read to go on? Section A Quiz 1. No; a constant change of +1 in corresponds to different changes in.. Yes; a constant change of +1 in corresponds to a constant change of  in.. intercept:  = 16  () = 16 = 16 = 16 = 88 intercept:  = 16 ()  = 16  = = 16  =  challenge and etend a. = 1 = 1 = 6 = 6  = gal = 6 1 = = 6 6 =. intercept: = 18 () + 6 = = = =  6 intercept: = () = = = = Holt McDougal Algebra 1
24 . = = 6. intercept: + = + = = = = 1  Rain Gauge intercept: + = () + = = 1. =  + =  1 = = 1 = 7 The midpoint is at (  1, 7 ) 11. d = ÇÇÇÇÇÇÇÇÇÇ ( (  1 d = ÇÇÇÇÇÇÇÇÇ (1  ) + (186) d = ÇÇÇÇ d = ÇÇÇÇ d = ÇÇ d = 1 d = 1 1 = 1 The distance is 1 ft. 1. no 1. es; 1 Rain (in.) m = in./h. in./h 1 Time (h).1 in./h. in./h. in./h = = 1. =. A slope of. means peppers cost $./pound. 8. m = = = 8 = A slope of means the speed of the car is ft/s. 9. m = = = 18  = 6 A slope of 6 means the temperature decreases at a rate of 6 F/mi. 6 slopeintercept form check it out! 1a. Plot (, ). Count units up and 1 unit right and plot another point. Draw the line connecting  the two points.  b. Plot (, 1). Count units down  and units right and plot another point. Draw the line connecting the two points.  a. = m + b = 11 b. = m + b = c. = m + b 1 = 8() + b 1 =  + b + + = b = 8  a. = is in the form = m + b. Plot (, ). Count units up and units right and plot another point. Draw the line connecting  the two points Holt McDougal Algebra 1
25 b. 6 + = = = =  + Plot (, ). Count units down and 1 unit right and plot another point. Draw the line connecting the two points.  c. =  is in the form = m + b. Plot (, ). Count units up and 1 unit right and plot another point. Draw the line connecting  the two points.  a. An equation is = b. The intercept is. This is the cost for the deposit. The slope is 18. This is the cost per person. c. = 18 + = 18() + = 8 The cost of catering an event for guests is $8. think and discuss 1. (, b). (, .7). 1. Plot the point (, b). Graphing the Line Described b = m + b. Find a second point on the line b using the slope m to move horizontall and verticall from (, b).. Draw the line connecting the two points.. Plot (, 1). Count units up and 1 unit right and plot another point. Draw the line connecting the two points Plot (, ). Count units down  and 1 unit right and plot another point. Draw the line connecting the two points. . intercept =   m = 1  () m = = 1 = m + b = 1  =  7. = m + b =  =  9. = m + b  = (1) + b  =  + b = b = = m + b = = m + b 7 = () + b 7 = 1 + b = b =  1. =  6 is in the form = m + b. Plot (, 6). Count units up  and units right and plot another   point. Draw the line connecting the two points. eercises guided practice 1. Plot (, ). Count 1 unit up and units right and plot another  point. Draw the line connecting  the two points. . Plot (,.). Count. units up and 1 unit right and plot another point. Draw the line connecting the two points = = () = 1( + 1) = Plot (, 1). Count units up and 1 unit right and plot another point. Draw the line connecting the two points. 17 Holt McDougal Algebra 1
26 1. + =   =  + Plot (, ). Count units down and 1 unit right and plot another point. Draw the line connecting the two points.  1a. An equation is = b. The intercept is 1. This is the distance she has alread biked. The slope is 18. This is Helen s speed. c. = = 18() + 1 = 6 Helen will have biked 6 mi after hours. practice and problem solving 1. Plot (, 7). Count 1 unit up and 8 units right and plot another 6 point. Draw the line connecting the two points Plot (, ). Count 6 units down and 1 unit right and plot another   point. Draw the line connecting the two points Plot (, ). Count 1 unit up and 1 unit right and plot another point.  Draw the line connecting the two points Plot (, 6). Count units down 6 and units right and plot another point. Draw the line connecting the two points intercept = m =   m =  = 1 = m + b = 1 + = = m + b =  9. = m + b = = m + b =  1 (6) + b =  + b = b = = = m + b 8 = (6) + b 8 = b = 8 = 1 + Plot (, ). Count 1 unit up and units right and plot another point. Draw the line connecting the two points. + = =  + Plot (, ). Count units down and units right and plot another point. Draw the line connecting the two points.. + = = Plot (, 8). Count units down 8 and 1 unit right and plot another point. Draw the line connecting the two points.  6a. An equation is = b. The intercept is 17. This is the cost of the enrollment fee. The slope is. This is the monthl cost for the health club. c. = + 17 = (1) + 17 = 9 The cost for a one ear membership is $9. 18 Holt McDougal Algebra 1
27 7a b. intercept = 1 = m + b 181 m =  1 m = 1 = = possible 9. possible . Impossible; lines with the same slope are parallel and therefore cannot intersect. 1. Impossible; if a linear function does not have a intercept, then its graph does not intersect the ais. The ais is vertical so onl lines that do not intersect the ais are also vertical. But vertical lines cannot be graphs of functions. All nonvertical lines will intersect the ais, so ever linear function will have a intercept.. B; the intercept is 1 and the slope is 1.. C; the intercept is 1 and the slope is 1.. A; the intercept is 1 and the slope is.. Possible answer: = ; no; because it has an undefined slope and no intercept. 6a. b. = c. The intercept is. This is the distance from Sam s house to Ricardo s house. The slope is 1. This is the bos walking speed. TEST PREP 7. B; The intercept of = 1  is . Since + () = 8, (, ) is on + = 8, so it is the intercept. 8. J; First subtract from both sides to isolate . Then multipl both sides b 1 to get rid of the minus sign. 9. B; Since () + =, (, ) is on + =. So + = has a intercept of.. 6 = = = = = + The slope is. 1. Find the slope:  9 = = = = 11 The slope is 1. = m + b = 1 + () = 1  challenge and etend. A + B = C A A B = A + C B = + C A B B =  A + C B B The slope is  A. The intercept is C B B.. n + = n + = n + = = n + n =  ( n ) = () n = 6 Find the intercept: 8  = 6 8()  = 6  = = 6  =  The intercept is .. ; An number minus is the number itself; ; Addition Propert of Equalit (Add b to both sides.) 19 Holt McDougal Algebra 1
28 7 POINTSLOPE FORM Check it out! 1a.  1 = m( = (  1 ) a.   a. m =  1 b.  1 = m( () = (  ) + = (  ) b = =  6 =  1 = m( = (  6)  = =  1 b. m = ()  1 = 11 = 1 = 61 = m( = 6(  )  1 = = 68. m = = = =  1 = (  )  1 = = + 9 intercept: = = = intercept: = () + 9 = + 9 = 9. m =  1 = = =.  1 = m( =.(  1)  8. = =. + 6 =. + 6 =.(1) + 6 =. The cost of an ad that is 1 lines long is $.. think and discuss 1. Both are based on the slope and a point. Slopeint.: uses the point that contains the int.: pointslope: can use an point.. Pointslope: when ou know the slope and a point; Slopeint.: when ou know the slope and the int.. If ou know two points on the line: Use the two points in the slope formula to find the slope. Then use the slope and one of the points to write the equation in pointslope form. eercises guided practice = m( (6) = 1 (  ) + 6 = 1 (  ).  1 = m( (7) = (  ) + 7 = (  ) =  1 (  ())  8 = = = (  1)  1 = =  1 Writing the Equation of a Line If ou know the slope and intercept: If the slope is m and the intercept is b, then the equation is = m + b. If ou know the slope and a point on the line: Use the slope and the point to write the equation in pointslope form = m( = (  1) 1 Holt McDougal Algebra 1
29 9. m = =   () =  = 1  = 1 (  ())  = =  1. m = = = = =  1 (  1)  = = m = = = 8 = =  1 (  )  = = m = =  () =  = (  )  = + + = + 1. m = = = = 1  = 1(  )  = =  intercept: = = intercept: =  =  1. m = = (1) = 1  =  = ( (1))  = = + 1 intercept: = = =  intercept: = () + 1 = 1 1. m = = 99 = =  9 = (  )  9 = = + intercept: = = = 1 intercept: = () + = 16. m = = = 1 = 11 = m( = 1 (  1)  6 = = 1 + = 1 + = 1 () + = 9 The depth of the oil after half an hour is 9 ft. practice and problem solving = m( = (  (1)) 9  = 9 ( + 1) = m( = 8(  1) = m( () = (  ) + = (  ) () =  (  1).  1 = (  (1)) 7 + = = = = = 6 (  9)  = = m= = =  1 = = 1 (  7) 7 11 Holt McDougal Algebra 1
30  8 = = m = = = =  11 (  )  7 = = m =  1 = ( 1) =  =   = ( (1))  = = m = = = 6 =  = (  ) =  6. m = = = 1  = 1 (  ) = m =  1 = 1 ( ) ( 1) = 1 7 =  1 = (  6)  1 = =  intercept: = = intercept: =  = . m = = = 19 =   =  (  )  = = intercept: = =   = 6 intercept: =  ( ) + 8 = 8. m = = = 96 =  6 = (  ())  6 = = + 9 intercept: = = = 6 intercept: = ( ) + 9 = 9. =  +,6 = () +,6 = 1, gal. m =  1 = =  = = m( =  (  )  6 = = = =  1 (6) + 1 = The boiling point of water at 6 feet is F. 6a. m = = 18.1 = = = .9 (  ) 1. = = b. .9; the change in the amount in dollars remaining on the card after each download c. ; the initial amount in dollars on the card d. $1..9 = 16 songs Holt McDougal Algebra 1
31 . Alwas 1. Never. Sometimes a.  1 = m( =. (  ) b =. (  )  11 = = 6 inches c. From :1 to 6: is. hours =. (.  )  11 = = 16.6 inches. Possible answer: + =  (  ). Possible answer:  = (  ) 6. Possible answer:  1 = (  ) Student A is incorrect. Student A incorrectl wrote  () as  instead of +.. Possible answer: When ou know a point and the slope, ou can immediatel use pointslope form. When ou know two points, first use them to find the slope. Then use the pointslope form, just like in the first case. 1. Possible answer: Linear equations that describe vertical lines cannot be written in pointslope form because the have an undefined slope. All nonvertical lines represent functions, and the can all be written in pointslope form. a. SAT Scores Mean combined score Years since 198 b. Possible answer: slope: 1.; intercept: 99; = c. intercept: mean score in 198 slope: number of points b which the mean score is increasing each ear a. (, 1) and (6, 8) b. m =  1 = =  = = m( =  (  6)  8 = = c. The total time to reach Sharon s house occurs when the number of blocks to Sharon s house is. So substitute for. = = =  ( 1) =  ) 18 = Stephen takes 18 minutes to reach Sharon s house. TEST PREP ( . D; Substituting the slope and point into the slopepoint formula and simplifing gives D.. H; The slope between the two points is  so the answer must be F or H. B using the slopepoint formula and rearranging into the slopeintercept form, ou get =  + 1, so the intercept is 1. challenge and etend 6. + = = = = The intercept is. m = = = The slope is = = The slope is = m( =  (  )  1 = = Holt McDougal Algebra 1
32 8. m = ( = ) = =  1 = m( = ( )  1 = (  )  1 = = (  1 ) Line of Best Fit Check it out! 1. 8 = = = : () + () + () + () = 16; =  + 8: () + () + (1) + () = ; = is better. a.. strong positive correlation; likel causeandeffect (more education often contributes to higher earnings) Think and discuss 1.. Possible answer: rvalue Scatter Plot Description of Correlation strong negative weak negative none weak strong positive positive Eercises GUIDED PRACTICE 1. residual. correlation coefficient. = + 1: (1) + (1) + (1) + () = 19; =  1: (1) + () + () + () = ; = + 1 is better. a A B C D E b. Slope: for each book read, student s average will increase 1.7 points; int.: a student who reads books will have an average of 7.. c. 1.7(1) , or b. Slope: cost is $./d; int.: $6.8 is added to the cost of ever ball of arn. c. = 1; =.(1) = $ ; ver well (r .88) ; ver well (r .91) 6. strong negative correlation; likel causeandeffect (more time plaing video games often contributes to lower test averages) PRACTICE AND PROBLEM SOLVING 7. =  + 8: (1) + () + (1) + (1) = 7; = : () + () + () + (1) = 9; = is better. 1 Holt McDougal Algebra 1
33 8a. 1a b. Slope: the famil will use. fewer gal/mo for each 1 F increase in mean temp.; int.: the famil will use 181 gal in a month when the mean temp is F. c. .() gal ; r.96 b. Slope: each ear there will be 11.6 more visitors than the previous ear; int: there were 16 visitors in. c. Yes; r.96, which is ver close to 1. d. No; the passage of time likel does not cause changes in the number of visitors. 16a.. + ; ver well (r.9) 1. moderatel strong positive correlation; unlikel causeandeffect (time spent on one activit in week 1 probabl does not affect time spent on the other activit in week ) ; .78 is closer to 1 than.6 is to Ever data point lies on the leastsquares line; residuals can be positive or negative, so their sum could be even when data points are not on the leastsquares line. 1a ; r.98 b. Slope: there will be $11.8 in sales for each visitor; int.: there will be $9 in sales if there are no visitors. (This could not actuall happen.) c. Yes; r.98, which is ver close to 1. However, predictions for small numbers of visitors might not be useful because of the neg. int. d. Yes; more visitors is likel to mean more mone spent in the gift shop. STANDARDIZED TEST PREP 17. Since the correlation is negative and the points do not form a straight line, choice B is correct b. Slope: a plaer will score.8 run for ever hit. c. int.: a plaer will score 1. runs if he has hits d. strong positive correlation; r.8, which is near 1. e..8(1) runs () + () + (1) + (1) + () = ; The correct choice is J ;.(89) ; 9 cases CHALLENGE AND EXTEND 19a. (1) + () + () + (1) + (1) + () + () + (1) = b. 8 = 8 8 = 1; 1 Holt McDougal Algebra 1
Analytic Geometry 300 UNIT 9 ANALYTIC GEOMETRY. An air traffi c controller uses algebra and geometry to help airplanes get from one point to another.
UNIT 9 Analtic Geometr An air traffi c controller uses algebra and geometr to help airplanes get from one point to another. 00 UNIT 9 ANALYTIC GEOMETRY Copright 00, K Inc. All rights reserved. This material
More informationChapter 4 ( 2, 2 ). Chapter 4 Opener. Section 4.1. Big Ideas Math Blue WorkedOut Solutions. Try It Yourself (p. 141) = = = = 15. The solution checks.
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