LESSON Answers for the lesson “Write Linear 4.1 …€¦ · 16. The given slope and y-intercept ... Answers for the lesson “Write Linear ... in Slope-Intercept Form ...
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43. No; the slope of the line is undefined, the equation is x 5 3, which is not in slope- intercept form.
44. Find the slope by substituting
the values: b 1 m 2 b } 1 2 0 5 m. The
y-intercept is when x 5 0, so the y-intercept is b. If you substitute (21, b 2 m) into the equation y 5 mx 1 b, you get b 2 m 5 2m 1 b which is a true statement.
Problem Solving
45. a. C 5 44m 1 48
b. $312
46. C 5 3.99e 1 1.49; $33.41
47. C 5 3h 1 30; $42
48. a. a 5 0.0037e 1 3
b. dependent variable: a, independent variable: e
c. Substitute 2 for e to get approximately 3.
49. a. x (yearssince 1970)
y(km2)
0 5.2
10 4.1
20 3.0
30 1.9
b.
100 20 30
6543210 !
"
The area of the glaciers changed 21.1 square kilometers between every 10 year interval.
c. y 5 20.11x 1 5.2; 20.11 km2
50. a. 81 million gal
b. y 5 130,000,000h
c. 0 h 3; water is only released for 3 hours after 10 A.M.
not intersect, they have the same slope, so they are parallel.
42. The three points lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see that all three points are on the
line y 5 3 } 4 x 1 1.
43. The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line.
44. The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line.
45. The three points do not lie on the same line. If you find the equation of the line between two of the points and then check to see that the third point is a solution, you can see they do not lie on the same line.
46. 7; find the equation of the line through (22, 3) and (2, 5) to be
y 5 1 } 2 x 1 4, then substitute 6 for
x to find k.
Problem Solving
47. 3 } 4 ft/yr; 6 ft
48. a. $2.95 b. $25.95
49. 115 min or 1 h 55 min; substitute 30 for m, 2 for x, and 85 for y into the equation y 5 mx 1 b to find b 5 25. Then substitute 3 for x into the equation y 5 30x 1 25 to solve for y.
The slope is the rate that the hurricane is traveling, the y-intercept represents the distance from the town at 12 P.M.
c. 1 A.M.; find the t-intercept to find the value of t when the distance to the town is 0; substitute 0 for d and solve for t; t 5 13, so you need to add 13 hours to 12 P.M. to get 1 A.M.
54. a. d 5 10t 1 60
b. The rate of change, 10 meters per second, represents the skater’s top racing speed; the initial value, 60, represents the distance the skater traveled from a stand-still to where he reached his top racing speed; d meters
5 1 10 meters } seconds 2 (x seconds)
1 60 meters.
c. 54 sec; the total distance is the length of the race track times 3 laps, 600 meters. If you substitute 600 for d, t 5 54.
2. Find the slope and substitute it for m in the equation y 2 y1 5 m(x 2 x1). Then pick one of the points and substitute the coordinates in for y1 and x1.
3. y 2 1 5 2(x 2 2)
4. y 2 5 5 2(x 2 3)
5. y 1 1 5 26(x 2 7)
6. y 1 1 5 22(x 2 5)
7. y 2 2 5 5(x 1 8)
8. y 2 6 5 3 } 2 (x 1 6)
9. y 1 3 5 29(x 1 11)
10. y 1 9 5 7 } 3 (x 1 3)
11. y 1 12 5 2 2 } 5 (x 2 5)
12. C
13. The form y 2 y1, so the left side should be y 2 (25) or y 1 5; y 1 5 5 22(x 2 1).
14.
!
!
"
y 5 3(x 1)
"
15.
"
"
!
"
y 3 2(x 2)
16.
!
!
"
"
y 1 3(x 6)
Answers for the lesson “Write Linear Equations in Point-Slope Form”
31. No; because the increase is not at a constant rate, the situation cannot be modeled by a linear equation.
32. Yes; because the rate is increas-ing at a constant rate, the situation can be modeled by a linear equa-tion. Sample answer:
y 2 1.2 5 1 } 5 (x 2 1)
33. No; because the increase is not at a constant rate, the situation cannot be modeled by a linear equation.
34. Yes; because the rate is decreas-ing at a constant rate, the situation can be modeled by a linear equa-tion. Sample answer: y 2 16 5 3(x 1 3)
35. 2; y 2 8 5 2(x 2 2) or y 2 6 5 2(x 2 4)
36. 23; y 2 3 5 3(x 2 4) or y 5 3(x 2 3)
Problem Solving
37. a. y 5 130x 1 530
b. $1570
38. Since the cost increases at a constant rate of $1714 per month, the situation can be modeled by a linear equation; $5950; $1714.
39. y 5 10,000x 1 67,000; $127,000
40. a. y 5 1.4x 1 30
b. 45.4 gal
41. a. Since the cost increases at a constant rate of $.49 per print, the situation can be modeled by a linear equation.
b. Sample answer: y 2 1.98 5 0.49(x 2 1)
c. $1.49 d. $1.79
42. a. y 5 2.45x 1 13.45
b. about 26.64 million metric tons
43. a. y 2 17.6 5 20.06(x 2 60)
b. 16.4 ft/sec
44. a. y 5 0.417391x 1 21.413
b. 1.59 billion lb; find the number of cans recycled per pound of aluminum in 2002 by substituting 30 for x to get about 33.9 cans per pound. Divide 53.8 billion aluminum cans by 33.9 cans per pound to find the number of pounds of aluminum.
4. Find the slope of the line then substitute the slope and one of the points into the point-slope form. Collect variables on one side and constants on the other side.
5–10. Sample answers are given.
5. 2x 1 2y 5 220, 3x 1 3y 5 230
6. x 1 2y 5 3, 10x 1 20y 5 30
7. x 2 2y 5 29, 22x 1 4y 5 18
8. 23x 2 4y 5 2, 26x 2 8y 5 4
9. 3x 2 y 5 24, 6x 2 2y 5 28
10. 2x 2 4y 5 5, 24x 1 8y 5 210
11. 2x 1 y 5 5
12. 23x 1 y 5 213
13. 2x 1 y 5 5
14. 4x 1 y 5 232
15. 3 } 2 x 1 y 5 210
16. 2 1 } 6 x 1 y 5 29
17. 2 } 3 x 1 y 5 2 4 } 3
18. 2x 1 y 5 7
19. 2 4 } 3 x 1 y 5 21
20. 24x 1 y 5 23
21. 2 1 } 2 x 1 y 5 1
22. y 5 22
23. y 5 2, x 5 3
24. y 5 23, x 5 25
25. y 5 3, x 5 21
26. y 5 3, x 5 5
27. y 5 4, x 5 21
28. y 5 22, x 5 26
29. (1, 24) was substituted incorrectly, 1 should be substituted for x and 24 substituted for y. A(1) 2 3(24) 5 5, A 1 12 5 5, A 5 27.
30. x } 6 1 y } 4 5 1; Sample answer:
I solved the equations 2x 1 3(0) 5 12 and 2(0) 1 3y 5 12 to find a, the x-intercept, and b, the y- intercept, respectively. Then I substituted the values of a and b into the general intercept form.
31. 4; 4x 1 3y 5 5
32. 1 } 2 ; 1 } 2 x 2 4y 5 21
33. 24; 2x 2 4y 5 10
Answers for the lesson “Write Linear Equations in Standard Form”
The n-intercept, 5, is the number of nights of boarding the dog at the kennel without any treats. The t-intercept, 20, is the number of treats that can be bought without boarding the dog for any nights.
41. a. 100 1 40s 5 1600
b.
2 0 4 6 8 10 12 14 16
40 35 30 25 20 15 10 5 0 #
$
c. Largerafts
Smallrafts
16 0
14 5
12 10
10 15
8 20
6 25
4 30
2 35
0 40
42. a. 0.75b 1 s 5 63
b. 18 subway rides; if you ride the bus 60 times, it costs (0.75)($60) 5 $45 without the pass. The pass costs $63, you need to spend $18 on subway rides, $18 4 $1 5 18.
29. Yes; the slope of the line through (4, 3) and (3, 21) is 4 and the slope of the line through (23, 3)
and (1, 2) is 2 1 } 4 . The slopes are
negative reciprocals, so the lines are perpendicular.
30. Sample answer: y 5 2x 1 1 and
y 5 2x 1 3; y 5 2 1 } 2 x 1 2
31. m 5 x1 2 x2 } y1 2 y2
Problem Solving
32. a. y 5 2x 1 8
b. y 5 22x 1 8
c. No; the slopes 22 and 2 are not negative reciprocals.
33. a. w1 5 200d 1 6000; w2 5 200d 1 6250
b. 12,000 lb; 12,250 lb
c. The graphs of the lines are parallel because they have the same slope, 200. The w-intercept of the second line is 250 more than the w-intercept of the first line.
34. Parallel: 2nd Street and Park
Street; the slope of both streets
is 2 } 3 . Since they have the same
slope, the streets are parallel. Perpendicular: 2nd Street and Sea
Street, Park Street and Sea Street;
the slope of Sea Street is 2 3 } 2 ,
which is the negative reciprocal
of 2 } 3 , the slope of 2nd Street and
Park Street. Since the slopes are negative reciprocals, the streets are perpendicular.
35. Different registration fees; because the lines are parallel, the rate of change, the monthly fee, for each must be equal. Therefore, the students paid different registration fees.
36. a. C 5 38.75m 1 49
b. C 5 38.75m 1 149
c. The graphs of the lines are parallel; they have the same slope, 38.75. The C-intercept of the second graph is 100 more than the C-intercept of the first graph.
b. The graphs of the lines are parallel; they have the same slope, 22.5. The y-intercept of the second line is 20 less then the y-intercept of the first line.
c. 20; 12; the x-intercepts show the number of months of nonuse it would take for the value of the gift card to be $0.
2. When data have a positive correlation, the dependent variable tends to increase as the independent variable increases. When data have a negative correlation, the dependent variable tends to decrease as the independent variable increases. When data have relatively no correlation where there is no apparent relationship between the independent variable and the dependent variable.
3. positive correlation
4. relatively no correlation
5. negative correlation
6. Sample answer: y 5 11.5x 2 0.28
0 1 2 3 4 5 6 7 9 8 !
"
0 10 20 30 40 50 60 70 80 90
100
7. Sample answer: y 5 24.5x 1 15.4
!2" !
"
8. C
9. The line does not have approximately half the data above it and half below it.
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10. The independent variable is x, not y; the dependent variable decreases as x increases.
11. Sample answer: The amount of time driving a car and the amount of gas left in the gas tank.
The growth rate of alligator 2 is slightly greater than the growth rate of alligator 1.
19. Sample answer: y 5 12.6x 1 32
20. a. Sample answer: y 5 1.2x 1 30
b. Sample answer: 1.2 min per day
c. No; it will continue through June and then start decreasing.
21. a. Sample answer: h 5 17.8y 1 93.6
b. Sample answer: m 5 0.78h 2 33.2
c. Sample answer: m 5 13.884y 1 39.808; the function models the amount of money, m, spent on the Internet as a function of the number of years, y, since 1999.
2. Extrapolation is finding an approximate value ouside the range of known values. Interpolation is finding an approximate value within the range of known values.
3. y 5 2.6x 1 2.3; 15.3
2 310 4 5 6 7 !
"
0
42
810
6
1214161820
4. y 5 8.2x 2 10.1; 30.9
0 2 4 6 8 10 !
"
010203040506070
5. y 5 10.7x 1 20; 127
2 310 4 5 6 !
"
0
2010
4050
30
6070
6. y 5 0.33x 1 0.22; 3.52
0 1 2 3 4 5 6 7 8 9 !
"
00.5
11.5
22.5
3
7. 2 2 } 3 8. 7 9. 216
10. 4 11. 1.5 12. 1.4
13. To find the zero of a function, substitute 0 for y, not x;
0 5 2.3x 2 2, 2 5 2.3x,
x 5 20
} 23 .
14. B
15. a and b were not substituted correctly; y 5 4.47x 1 23.1.
Answers for the lesson “Predict with Linear Models”
22. c. The slope, 20.2, is the change in the cost (in thousands of dollars per thousand miles) of a local car of the same model, make, and year, and the y-intercept, 19.7, is the predi-cated selling price (in thou-sands of dollars) for a local car of the same model, make, and year with a mileage of 0.
23. a.
0 5 10 15 20 25 !
"
0
2000
4000
6000
8000
0 5 10 15 20 25 !
"
0
10,000
20,000
30,000
40,000
There is relatively no correlation in either scatter plot.
b. No; because you cannot find a line of best fit for either correlation, you cannot use the mallard duck population to predict the total duck population.