MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the system of equations.

1) x1 - x2 + 3x3 = -8
2x1 + x3 = 0
x1 + 5x2 + x3 = 40
A) (8, 8, 0) B) (0, -8, -8) C) (-8, 0, 0) D) (0, 8, 0)
Answer: D

2) x1 + 3x2 + 2x3 = 11
4x2 + 9x3 = -12
x3 = -4
A) (1, 6, -4) B) (-4, 1, 6) C) (1, -4, 6) D) (-4, 6, 1)
Answer: A

3) x1 - x2 + 8x3 = -107
6x1 + x3 = 17
3x2 - 5x3 = 89
A) (-5, 8, 13) B) (5, -8, -13) C) (5, 8, -13) D) (-5, -8, 13)
Answer: C

4) 4x1 - x2 + 3x3 = 12
2x1 + 9x3 = -5
x1 + 4x2 + 6x3 = -32
A) (2, -7, 1) B) (2, 7, 1) C) (2, 7, -1) D) (2, -7, -1)
Answer: D

5) x1 + x2 + x3 = 6
x1 - x3 = -2
x2 + 3x3 = 11
A) (0, 1, 2) B) (-1, 2, -3) C) (1, 2, 3) D) No solution
Answer: C

6) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
5x1 + x2 + x3 = 11
A) (4, 2, 1) B) (4, 1, 2) C) (1, 2, 4) D) (1, 4, 2)
Answer: C

7) x1 - x2 + x3 = 8
x1 + x 2 + x 3 = 6
x1 + x2 - x3 = -12
A) (2, -1, 9) B) (2, -1, -9) C) (-2, -1, -9) D) (-2, -1, 9)
Answer: D

1
 8) 5x1 + 2x2 + x3 = -11
2x1 - 3x2 - x3 = 17
7x1 + x2 + 2x3 = -4
A) (3, 0, -4) B) (0, -6, 1) C) (0, 6, -1) D) (-3, 0, 4)
Answer: B

9) 7x1 + 7x2 + x3 = 1
x1 + 8x2 + 8x3 = 8
9x1 + x2 + 9x3 = 9
A) (0, 1, 0) B) (1, -1, 1) C) (-1, 1, 1) D) (0, 0, 1)
Answer: D

10) 2x1 + x2 =0
x1 - 3x2 + x3 = 0
3x1 + x2 - x3 = 0
A) (0, 0, 0) B) (0, 1, 0) C) No solution D) (1, 0, 0)
Answer: A

Determine whether the system is consistent.

11) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
5x1 + x2 + x3 = 11
A) Yes B) No
Answer: A

12) 5x1 + 2x2 + x3 = -11

2x1 - 3x2 - x3 = 17
7x1 + x2 + 2x3 = -4
A) Yes B) No
Answer: A

13) 4x1 - x2 + 3x3 = 12

2x1 + 9x3 = -5
x1 + 4x2 + 6x3 = -32
A) Yes B) No
Answer: A

14) 2x1 + x2 =0
x1 - 3x2 + x3 = 0
3x1 + x2 - x3 = 0
A) Yes B) No
Answer: A

2
 15) x1 + x2 + x3 = 6
x1 - x3 = -2
x2 + 3x3 = 11
A) No B) Yes
Answer: B

16) x1 - x2 + 3x3 = -11

-4x1 + 4x2 - 12x3 = -2
x1 + 3x2 + x3 = -17
A) Yes B) No
Answer: B

17) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
2x1 + 3x3 = 15
A) Yes B) No
Answer: B

18) x1 + 3x2 + 2x3 = 11

4x2 + 9x3 = -12
x1 + 7x2 + 11x3 = -11
A) Yes B) No
Answer: B

19) 5x1 + 2x2 + x3 = -11

2x1 - 3x2 - x3 = 17
7x1 - x2 = 12
A) No B) Yes
Answer: A

20) 5x2 + x4 = -21

x1 + x2 + 4x3 - x4 = 4
5x1 + x3 + 4x4 = 12
x1 + x2 + 6x3 =5
A) No B) Yes
Answer: B

Determine whether the matrix is in echelon form, reduced echelon form, or neither.
1 3 5 -7
21) 0 1 -4 -4
0 0 1 6

A) Reduced echelon form B) Echelon form C) Neither

Answer: B

3
 1 4 5 -7
22) 0 1 -4 -5
0 6 1 4

A) Echelon form B) Reduced echelon form C) Neither

Answer: C

1 4 5 -7
23) 3 1 -4 -6
0 4 1 3

A) Reduced echelon form B) Echelon form C) Neither

Answer: C

1 0 0 -7
24) 1 1 0 1
0 3 1 1

A) Echelon form B) Reduced echelon form C) Neither

Answer: C

1 4 1 -7
25) 0 1 -4 7
0 0 0 0

A) Neither B) Echelon form C) Reduced echelon form

Answer: B

1 0 5 -4
26) 0 1 -5 -2
0 0 0 0
0 0 0 0

A) Echelon form B) Reduced echelon form C) Neither

Answer: B

1 -5 -5 -5
27) 0 0 -2 3
0 0 0 -3
0 0 0 0

A) Neither B) Reduced echelon form C) Echelon form

Answer: C

4
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
28) Find the echelon form of the given matrix.
1 4 -2 3
-3 -11 9 -5
-2 4 -3 4
A) B) C) D)
1 4 -2 3 1 4 -2 3 1 4 -2 3 1 4 -2 3
0 1 3 4 0 1 3 4 0 1 3 4 0 1 3 4
0 12 -7 10 0 0 -43 -38 0 0 -43 0 0 0 -19 -2
Answer: B

29) Find the reduced echelon form of the given matrix.

1 4 -5 1 2
2 5 -4 -1 4
-3 -9 9 2 2
A) B)
1 4 -5 0 -2 1 0 0 0 14
0 1 -2 0 -4 0 1 0 0 -4
0 0 0 1 4 0 0 0 1 4
C) D)
1 4 -5 1 2 1 0 3 0 14
0 1 -2 1 0 0 1 -2 0 -4
0 0 0 1 4 0 0 0 1 4
Answer: D

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise
state that there is no solution.
30) 1 -5 -1
0 0 3

A) (-1, 3) B) x1 = -1 + 5x2 C) x1 = -1 + 5x2 D) No solution

x2 is free x2 = 3
x3 is free
Answer: D

1 2 -3 -9
31) 0 1 4 5
0 0 0 1

A) x1 = -19 + 11x3 B) No solution

x2 = 5 - 4x3
x3 = 1
C) x1 = -9 - 2x2 + 3x3 D) x1 = -19 + 11x3
x2 is free x2 = 5 - 4x3
x3 is free x3 is free
Answer: B

5
 1 2 -3 6
32) 0 1 4 -7
0 0 0 0

A) x1 = 6 -2x2 + 3x3 B) x1 = 6 - 2x2 + 3x3 C) x1 = 20 + 11x3 D) x1 = 20 + 11x3

x2 = -7 - 4x3 x2 is free x2 = -7 - 4x3 x2 = -7 - 4x3
x3 is free x3 is free x3 = 0 x3 is free
Answer: D

1 0 6 2
33) 0 1 -2 -3
0 0 0 0

A) x1 = 2 - 6x3 B) No solution C) x1 = 2 - 6x3 D) x1 = 2 - 6x3

x2 = -3 + 2x3 x2 = -3 + 2x3 x2 is free
x3 = 0 x3 is free 3 1
x3 = + x
2 2 2
Answer: C

1 4 -2 -3 1
34) 0 0 1 4 -4
-1 -4 -1 -9 11

A) x1 = - 4x2 +2x3 + 3 x4 + 1 B) x1 = -7 - 4x2 - 5x3

x2 is free x2 = -4 - 4x3
x3 = -4 - 4x4 x3 is free
x4 is free

C) x1 = -7 - 4x2 - 5x4 D) x1 = -7 - 4x2 - 5x4

x2 is free x2 is free
x3 = -4 - 4x4 x3 = -4 - 4x4
x4 = 0 x4 is free

Answer: D

6
 1 6 3 -1 2 6
35) 0 0 0 -4 3 4
0 0 0 0 -2 8

A) No solution B) x1 = -6x2 - 3x3 + x4 - 2x5 + 6

x2 is free
x3 is free
3
x4 = x -1
4 5
x5 = -4

C) x1 = -6x2 - 3x3 + 10 D) x1 = -6x2 - 3x3 + 10

x2 is free x2 is free
x3 is free x3 = -4
x4 = -4 3
x4 = x -1
x5 = -4 4 5
x5 = -4

Answer: C

Find the indicated vector.

36) Let u = -6 , v = 2 . Find u + v.
4 1
A) B) C) D)
-8 -2 -5 -4
3 3 6 5
Answer: D

37) Let u = 9 , v = 7 . Find u - v.

-8 2
A) B) C) D)
7 2 17 16
-15 -10 5 -6
Answer: B

38) Let u = -5 , v = 1 . Find v - u.

5 6
A) B) C) D)
11 6 -4 10
-4 1 11 5
Answer: B

39) Let u = 3 . Find 4u.

2
A) B) C) D)
-12 -12 12 12
-8 8 8 -8
Answer: C

7
 40) Let u = 8 . Find 3u.
-6
A) B) C) D)
24 -24 -24 24
-18 -18 18 18
Answer: A

41) Let u = -6 . Find -2u.

-7
A) B) C) D)
12 -12 -12 12
14 14 -14 -14
Answer: A

42) Let u = -7 . Find -5u.

5
A) B) C) D)
35 -35 35 -35
-25 -25 25 25
Answer: A

43) Let u = 2 , v = 7 . Find -5u + 2v.

1 -5
A) B) C) D)
-45 4 -24 -15
-8 -15 5 4
Answer: B

Display the indicated vector(s) on an xy-graph.

44) Let u = -3 and v = -2 . Display the vectors u, v, and u + v on the same axes.
5 2

8
 A) B)

C) D)

Answer: C

45) Let u = 5 Display the vector 2u using the given axes.

-4

9
 A) B)

C) D)

Answer: D

Solve the problem.

6 -5 -39
46) Let a 1 = 1 , a 2 = 1 , and b = -1 .
-1 6 22
Determine whether b can be written as a linear combination of a 1 and a 2 . In other words, determine whether
weights x1 and x2 exist, such that x1 a1 + x2 a2 = b. Determine the weights x1 and x2 if possible.
A) x1 = -4, x2 = 4 B) x1 = -4, x2 = 3 C) x1 = -3, x2 = 2 D) No solution
Answer: B

1 -3 2 -3
47) Let a 1 =2 , a 2 = -4 , a 3 = 1 , and b = 6 .
-3 1 6 -1
Determine whether b can be written as a linear combination of a 1 , a 2 , and a 3 . In other words, determine whether
weights x1 , x2 , and x3 exist, such that x1 a 1 + x2 a 2 + x3 a 3 = b. Determine the weights x1 , x2 , and x3 if
possible.
A) No solution B) x1 = -5, x2 = 0, x3 = 1
C) x1 = -2, x2 = -1, x3 = 6 D) x1 = 2, x2 = 1, x3 = - 1
Answer: A

10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

48) A company manufactures two products. For $1.00 worth of product A, the company spends $0.50 on materials,
$0.20 on labor, and $0.10 on overhead. For $1.00 worth of product B, the company spends $0.40 on materials,
$0.20 on labor, and $0.10 on overhead. Let

0.50 0.40
a = 0.20 and b = 0.20 .
0.10 0.10

Then a and b represent the "costs per dollar of income" for the two products.
Evaluate 500a + 100b and give an economic interpretation of the result.
290
Answer: 500a + 100b = 120
60

500a + 100b lists the various costs for producing $500 worth of product A and $100 worth of product B,
namely $290 for materials, $120 for labor, and $60 for overhead.

49) A company manufactures two products. For $1.00 worth of product A, the company spends $0.45 on materials,
$0.20 on labor, and $0.10 on overhead. For $1.00 worth of product B, the company spends $0.40 on materials,
$0.20 on labor, and $0.10 on overhead. Let

0.45 0.40
a = 0.20 and b = 0.20 .
0.10 0.10

Then a and b represent the "costs per dollar of income" for the two products.
Suppose the company manufactures x1 dollars worth of product A and x2 dollars worth of product B and that its
total costs for materials are $205, its total costs for labor are $100, and its total costs for overhead are $50.
Determine x1 and x2 , the dollars worth of each product produced. Include a vector equation as part of your
solution.
205
Answer: x1 a + x2 b = 100
50
or

0.45 0.40 205

x1 0.20 + x2 0.20 = 100
0.10 0.10 50

x1 = 100, x2 = 400

11
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Compute the product or state that it is undefined.

9
50) [-7 2 7] 0
-3
A) B) C) D)
-63 [-84] [-63 0 -21] [435]
0
-21
Answer: B

5
51) -1 1 -2 9
-8 2 -6
9
A) B) C) D)
[-14 -76] -1 1 -2 -14 -76
-8 2 -6 -76 -14
59 9
Answer: C

52)
3 -7
-4
8 -4
5
-4 1

A) Undefined B) C) D)
-47 -12 40 -12 -35
-52 28 -20 -32 -20
21 -4 1 16 5
Answer: B

53)
-4 8 1
1 5 -5
2 8 7
A) Undefined B) C) D)
-4 8 5 39 4
-5 -25 -30
14 56 70
Answer: A

12
Write the system as a vector equation or matrix equation as indicated.
54) Write the following system as a vector equation involving a linear combination of vectors.
5x1 - 2x2 - x3 = 2
4x1 + 3x3 = -1

5 4 2
A) x1 2 + x2 0 = -1 B) x1 5 + x2 -2 + x3 -1 = 2
4 0 3 -1
-1 3 0
x1 x1 x1
2
C) 5 x2 - 2 x2 - x2 = -1 D) x1 5 + x2 -2 + x3 1 = 2
4 1 3 -1
x3 x3 x3 0
Answer: B

55) Write the following system as a matrix equation involving the product of a matrix and a vector on the left side and a
vector on the right side.
3x1 + x2 - 2x3 = -4
2x1 - 2x2 =1
x1
3
x1 x2 x3
A) 1 = -4 B) 3 1 -2 x2 = -4
2 -2 0 1 2 2 1 1
-2 x3
x1
3 2 x1
C) 3 1 -2 x = -4 D) = -4
2 1 -2
2 -2 0 1 x2 1
x3 -2 0
Answer: C

Solve the problem.

b1
1 -3 2
56) Let A = -2 5 -1 and b = b2 .
3 -4 -7 b3

Determine if the equation Ax = b is consistent for all possible b1 , b2 , b3 . If the equation is not consistent for all
possible b1 , b2 , b3 , give a description of the set of all b for which the equation is consistent (i.e., a condition
which must be satisfied by b1 , b2 , b3 ).
A) Equation is consistent for all possible b1 , b2 , b3 .
B) Equation is consistent for all b1 , b2 , b3 satisfying 2b1 + b2 = 0.
C) Equation is consistent for all b1 , b2 , b3 satisfying 7b1 + 5b2 + b3 = 0.
D) Equation is consistent for all b1 , b2 , b3 satisfying -3b1 + b3 = 0.
Answer: A

13
 b1
1 -3 2
57) Let A = -2 5 -1 and b = b2 .
3 -3 -12 b3

Determine if the equation Ax = b is consistent for all possible b1 , b2 , b3 . If the equation is not consistent for all
possible b1 , b2 , b3 , give a description of the set of all b for which the equation is consistent (i.e., a condition
which must be satisfied by b1 , b2 , b3 ).
A) Equation is consistent for all b1 , b2 , b3 satisfying -3b1 + b3 = 0.
B) Equation is consistent for all b1 , b2 , b3 satisfying -b1 + b2 + b3 = 0.
C) Equation is consistent for all b1 , b2 , b3 satisfying 9b1 + 6b2 + b3 = 0.
D) Equation is consistent for all possible b1 , b2 , b3 .
Answer: C

58) Find the general solution of the simple homogeneous "system" below, which consists of a single linear equation.
Give your answer as a linear combination of vectors. Let x2 and x3 be free variables.
-2x1 - 8x2 + 16x3 = 0
A) B)
x1 x1 x1 x1
4 -8
x 2 = x2 1 + x3 0 (with x2, x3 free) x2 = -4 x2 - 8 x2 (with x2 , x3 free)
x3 0 1 x3 x3 x3
C) D)
x1 x1
-4 8 -4 8
x 2 = x2 0 + x 3 1 (with x2, x3 free) x 2 = x2 1 + x3 0 (with x2, x3 free)
x3 1 0 x3 0 1
Answer: D

59) Find the general solution of the homogeneous system below. Give your answer as a vector.
x1 + 2x2 - 3x3 = 0
4x1 + 7x2 - 9x3 = 0
-x1 - 4x2 + 9x3 = 0
A) B) C) D)
x1 x1 x1 x1
-3 3 -3 -3
x 2 = x3 3 x2 = x3 -3 x 2 = x3 3 x2 = 3
x3 0 x3 1 x3 1 x3 1
Answer: C

14
60) Describe all solutions of Ax = b, where
2 -5 3 -5
A = -2 6 -5 and b = 2 .
-4 7 0 19

Describe the general solution in parametric vector form.

A) B)
x1 x1
-10 7/2 -5 -1
x2 = -3 + x3 2 x2 = -3 + x3 2
x3 0 1 x3 0 1
C) D)
x1 x1
7/2 -10 -10 7/2
x2 = 2 + x3 -3 x2 = -3 + x3 2
x3 1 0 x3 0 0
Answer: A

61) Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A).
Sector E sells 70% of its output to M and 30% to A.
Sector M sells 30% of its output to E, 50% to A, and retains the rest.
Sector A sells 15% of its output to E, 30% to M, and retains the rest.

Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by
pe, pm , and pa, respectively. If possible, find equilibrium prices that make each sector's income match its expenditures.
Find the general solution as a vector, with pa free.
A) B)
pe 0.607 pa pe 0.465 pa
pm = 0.481 pa pm = 0.593 pa
pa pa pa pa
C) D)
pe 0.356 pa pe 0.308 pa
pm = 0.686 pa pm = 0.716 pa
pa pa pa pa
Answer: C

15
62) The network in the figure shows the traffic flow (in vehicles per hour) over several one-way streets in the
downtown area of a certain city during a typical lunch time. Determine the general flow pattern for the network.
In other words, find the general solution of the system of equations that describes the flow. In your general solution let
x4 be free.

A) x1 = 600 - x4 B) x1 = 500 + x4 C) x1 = 600 + x5 D) x1 = 600 - x4

x2 = 400 - x4 x2 = 400 - x4 x2 = 400 - x5 x2 = 400 + x4
x3 = 300 + x4 x3 = 300 - x4 x3 = 300 - x5 x3 = 300 - x4
x4 is free x4 is free x4 = 300 x4 is free
x5 = 300 x5 = 200 x5 is free x5 = 300
Answer: D

1 -3 2
63) Let v1 = -3 , v2 = 8 , v3 = -2 .
-4 4 6

Determine if the set {v1 , v2 , v3 } is linearly independent.

A) Yes B) No
Answer: A

-2 1 4
64) Determine if the columns of the matrix A = 4 0 -4 are linearly independent.
2 4 6
A) No B) Yes
Answer: A

65) For what values of h are the given vectors linearly independent?
-1 -4
1 , 4
6 h
A) Vectors are linearly independent for h = 24 B) Vectors are linearly dependent for all h
C) Vectors are linearly independent for all h D) Vectors are linearly independent for h 24
Answer: D

16
66) For what values of h are the given vectors linearly dependent?
-1 5 5 -20
4 , 2 , -3 , 12
6 -3 5 h
A) Vectors are linearly independent for all h B) Vectors are linearly dependent for h -20
C) Vectors are linearly dependent for h = -20 D) Vectors are linearly dependent for all h
Answer: D

2
67) Let A = 2 8 -2 and u = -1 .
3 -5 -3
1
Define a transformation T: -> 2 by T(x) = Ax. Find T(u), the image of u under the transformation T.
3
A) B) C) D)
10 -6 16 4 -8 -2
-3 8 5 6 5 -3
-5
Answer: B

68) Let T: 2 -> 2 be a linear transformation that maps u = -3 into -15 and maps v = 6 into 24 .
6 6 -6 -12
Use the fact that T is linear to find the image of 3u + v.
A) B) C) D)
9 -3 -21 27
-6 12 6 -18
Answer: C

1 -3 0 2
69) Let A = -4 0 2 and b = 6 .
2 -5 -3 0
Define a transformation T: 3 -> 3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of the
transformation T.
A) B)
b is not in the range of the transformation T. -1
-1
0
C) D)
-1 -1
-1 1
1 -1
Answer: C

17
 1 -3 2 -5
70) Let A = -3 4 -1 and b = 2 .
2 -5 3 -4
Define a transformation T: 3 -> 3 by T(x) = Ax.
If possible, find a vector x whose image under T is b. Otherwise, state that b is not in the range of the
transformation T.
A) B)
-10 b is not in the range of the transformation T.
-5
-5
C) D)
2 4
2 0
0 -4
Answer: B

Describe geometrically the effect of the transformation T.

71) Let A = 1 0 .
2 1
Define a transformation T by T(x) = Ax.
A) Horizontal shear B) Vertical shear
C) Projection onto x1 -axis D) Projection onto x2 -axis
Answer: B

0 0 0
72) Let A = 0 1 0 .
0 0 1
Define a transformation T by T(x) = Ax.
A) Projection onto the x2 -axis B) Projection onto the x2 x3 -plane
C) Vertical shear D) Horizontal shear
Answer: B

18
Solve the problem.
1 0 0 1 0 0
73) The columns of I3 = 0 1 0 are e 1 = 0 , e 2 = 1 , e 3 = 0 .
0 0 1 0 0 1

Suppose that T is a linear transformation from 3 into 2 such that

T( e 1 ) = 3 , T( e ) = 2 , and T( e ) = -4 .
2 3
-2 0 1

x1
Find a formula for the image of an arbitrary x = x2 in 3 .
x3
A) B)
x1 x1
3x1 - 2x2 3x1 + 2x2 - 4x3
T x2 = T x2 =
2x1 -2x1 + x3
x3 x3
C) D)
x1 3x1 - 2x2 x1 3x1 + 2x2 - 4x3
T x2 = 2x1 T x2 = 2x1
x3 4x2 + x3 x3 -2x1 + x3
Answer: B

Find the standard matrix of the linear transformation T.

7
74) T: 2 -> 2 rotates points (about the origin) through radians (with counterclockwise rotation for a positive
4
angle).
A) B) C) D)
2 2 3 3 1 1 2 2
- -
2 2 3 3 -1 1 2 2
2 2 3 3 2 2
- - -
2 2 3 3 2 2
Answer: D

75) T: 2 -> 2 first performs a vertical shear that maps e 1 into e 1 + 3e 2 , but leaves the vector e 2 unchanged, then
reflects the result through the horizontal x1 -axis.
A) B) C) D)
-1 -3 1 0 1 3 -1 0
0 1 -3 -1 0 -1 3 -1
Answer: B

19
Determine whether the linear transformation T is one-to-one and whether it maps as specified.
76) Let T be the linear transformation whose standard matrix is
1 -2 3
A = -1 3 -4 .
-5 5 -6
Determine whether the linear transformation T is one-to-one and whether it maps 3 onto 3 .
A) One-to-one; not onto 3 B) Not one-to-one; not onto 3
C) One-to-one; onto 3 D) Not one-to-one; onto 3
Answer: C

77) T(x1 , x2 , x3 ) = (-2x2 - 2x3 , -2x1 + 9x2 + 5x3 , -x1 - 2x3 , 3x2 + 3x3 )
Determine whether the linear transformation T is one-to-one and whether it maps 3 onto 4 .
A) Not one-to-one; onto 4 B) Not one-to-one; not onto 4
C) One-to-one; not onto 4 D) One-to-one; onto 4
Answer: B

Solve the problem.

78) The table shows the amount (in g) of protein, carbohydrate, and fat supplied by one unit (100 g) of three different foods.

Food 1 Food 2 Food 3

Protein 15 35 25
Carbohydrate 45 30 50
Fat 6 4 1

Betty would like to prepare a meal using some combination of these three foods. She would like the meal to contain 15 g
of protein, 25 g of carbohydrate, and 3 g of fat. How many units of each food should she use so that the meal will
contain the desired amounts of protein, carbohydrate, and fat? Round to 3 decimal places.
A) 0.326 units of Food 1, 0.247 units of Food 2, 0.059 units of Food 3
B) 0.302 units of Food 1, 0.238 units of Food 2, 0.085 units of Food 3
C) 0.280 units of Food 1, 0.192 units of Food 2, 0.164 units of Food 3
D) 0.360 units of Food 1, 0.204 units of Food 2, 0.055 units of Food 3
Answer: A

79) The population of a city in 2000 was 400,000 while the population of the suburbs of that city in 2000 was
800,000.
Suppose that demographic studies show that each year about 5% of the city's population moves to the suburbs (and
95% stays in the city), while 4% of the suburban population moves to the city (and 96% remains in the suburbs).
Compute the population of the city and of the suburbs in the year 2002. For simplicity, ignore other influences on the
population such as births, deaths, and migration into and out of the city/suburban region.
A) City: 412,000 B) City: 361,000 C) City: 361,000 D) City: 422,920
Suburbs: 788,000 Suburbs: 839,000 Suburbs: 737,280 Suburbs: 777,080
Answer: D

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