Deck 9: Systems of Equations and Inequalities

Question 1

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Answer 1

Question 2

A stationary company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $1.1, and a regular notebook, which sells for $0.85. The production cost is $1.00 for each deluxe notebook and $0.75 for each regular notebook. The company has the facilities to manufacture between 2,000 and 3,000 deluxe and between 3,000 and 6,000 regular notebooks, but not more than 7,000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production cost?

A)2,000 deluxe and 5,000 regular
B)3,000 deluxe and 4,000 regular
C)2,300 deluxe and 4,000 regular
D)2,500 deluxe and 4,000 regular
E)3,000 deluxe and 5,000 regular

Answer 2

3,000 deluxe and 4,000 regular

Question 3

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Answer 3

Question 4

Solve the system.

A)
B)
C)
D)
E)

Question 5

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 6

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 7

A manufacturer of cell phones makes a profit of $23 on a deluxe model and $31 on a standard model. The company wishes to produce at least 80 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not exceed 240 phones. How many of each type should be produced daily in order to maximize the profit?

A)31 standard and 80 deluxe
B)80 standard and 160 deluxe
C)160 standard and 80 deluxe
D)200 standard and 40 deluxe
E)160 standard and 60 deluxe

Question 8

Sketch the region R determined by the given constraints, and label its vertices. Find the maximum value of C on R.

A)maximum of 20 at
B)maximum of 24 at
C)maximum of 0 at
D)maximum of 22 at
E)maximum of 18 at

Question 9

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Question 10

Find a system of inequalities for the graph shown.

A)
B)
C)
D)
E)

Question 11

Use matrices to solve the system.

A)
B)
C)
D)The system is dependent
E)No solution

Question 12

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent.
E)The equations are dependent.

Question 13

Use the method of substitution to solve the system.

A)( - 4, -21 )
B)( - 4, -21 ), ( 9, 44 )
C)( - 4, -21 ), ( 8, 44 )
D)( 9, 44 )
E)no solution

Question 14

Use matrices to solve the system.

A)
B)
C)
D)The equations are dependent
E)No solution

Question 15

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 16

A small firm manufactures bookshelves and desks for microcomputers. For each product it is necessary to use a table saw and a power router. To manufacture each bookshelf, the saw must be used for hour and the router for 1 hour. A desk requires the use of each machine for 2 hours. The profits are $20 per bookshelf and $50 per desk. If the saw can be used for 8 hours per day and the router for 12 hours per day, how many bookshelves and desks should be manufactured each day to maximize the profit?

A)4 bookshelves and 2 desks
B)2 bookshelves and 2 desks
C)2 bookshelves and 4 desks
D)0 bookshelves and 4 desks
E)2 bookshelves and 0 desks

Question 17

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 18

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 19

Three average monthly low temperatures for Detroit are listed in the table.
Let correspond to January, to February, . . . , and to December. Determine a quadratic function that interpolates the data - that is, determine the constants a, b, and c such that , and .

A)
B)
C)
D)
E)

Question 20

Use the method of substitution to solve the system.

A)( 0, 2 )
B)( 4, 0 )
C)( 4, 0 ), ( 0, 2 )
D)( 4, 1 ), ( 0, 2 )
E)no solution

Question 21

Use the method of substitution to solve the system.

A)( - 7, -18 )
B)( 9, 14 )
C)( - 7, -18 ), ( 9, 14 )
D)( - 7, -18 ), ( 8, 14 )
E)no solution

Question 22

A small firm manufactures bookshelves and desks for microcomputers. For each product it is necessary to use a table saw and a power router. To manufacture each bookshelf, the saw must be used for hour and the router for 1 hour. A desk requires the use of each machine for 2 hours. The profits are $20 per bookshelf and $50 per desk. If the saw can be used for 8 hours per day and the router for 12 hours per day, how many bookshelves and desks should be manufactured each day to maximize the profit?

A)8 bookshelves and 8 desks
B)0 bookshelves and 1 desks
C)8 bookshelves and 0 desks
D)8 bookshelves and 1 desks
E)1 bookshelves and 8 desks

Question 23

Use Cramer's rule, whenever possible, to solve the system.

A)
B)
C)
D)The equations are dependent.
E)The system is inconsistent.

Question 24

Use matrices to solve the system.

A)
B)
C)
D)The system is dependent
E)No solution

Question 25

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 26

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 27

A stationary company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $1.25, and a regular notebook, which sells for $0.9. The production cost is $1.00 for each deluxe notebook and $0.75 for each regular notebook. The company has the facilities to manufacture between 2,000 and 3,000 deluxe and between 3,000 and 6,000 regular notebooks, but not more than 8,000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production cost?

A)2,500 deluxe and 5,000 regular
B)2,000 deluxe and 6,000 regular
C)2,300 deluxe and 5,000 regular
D)3,000 deluxe and 5,000 regular
E)3,000 deluxe and 6,000 regular

Question 28

Find the determinant of the matrix.

A)15,232.88
B)3,046.58
C)22,849.32
D)7,616.44
E)30,465.76

Question 29

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Question 30

Find the determinant.

A)
B)
C)
D)
E)

Question 31

A manufacturer of cell phones makes a profit of $26 on a deluxe model and $30 on a standard model. The company wishes to produce at least 80 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not exceed 240 phones. How many of each type should be produced daily in order to maximize the profit?

A)160 standard and 60 deluxe
B)30 standard and 80 deluxe
C)80 standard and 160 deluxe
D)200 standard and 40 deluxe
E)160 standard and 80 deluxe

Question 32

Let be the identity matrix of order 2, and let . Find the polynomial for the given matrix A. (In the study of matrices, is the characteristic polynomial of A, and the zeros of are the characteristic values (eigenvalues) of A.)

A)
B)
C)
D)
E)

Question 33

Use matrices to solve the system.

A)
B)
C)
D)The equations are dependent
E)No solution

Question 34

Find a system of inequalities for the graph shown.

A)
B)
C)
D)
E)

Question 35

Sketch the region R determined by the given constraints, and label its vertices. Find the maximum value of C on R.

A)maximum of 44 at
B)maximum of 40 at
C)maximum of 42 at
D)maximum of 46 at
E)maximum of 0 at

Question 36

Solve the system.

A)
B)
C)
D)
E)

Question 37

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 38

Use the method of substitution to solve the system.

A)( 4, 0 ), ( -32, 6 )
B)( -32, 6 )
C)( 4, 1 ), ( -32, 6 )
D)( 4, 0 )
E)no solution

Question 39

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Question 40

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 41

Let be the identity matrix of order 2, and let . Find the polynomial for the given matrix A. (In the study of matrices, is the characteristic polynomial of A, and the zeros of are the characteristic values (eigenvalues) of A.)

A)
B)
C)
D)
E)

Question 42

Sketch the region R determined by the given constraints, and label its vertices. Find the maximum value of C on R.

A)maximum of 12 at
B)maximum of 14 at
C)maximum of 16 at
D)maximum of 0 at
E)maximum of 18 at

Question 43

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent.
E)The equations are dependent.

Question 44

Use matrices to solve the system.

A)
B)
C)
D)The equations are dependent
E)No solution

Question 45

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 46

Use Cramer's rule, whenever possible, to solve the system.

A)
B)
C)
D)The equations are dependent.
E)The system is inconsistent.

Question 47

Solve the system.

A)
B)
C)
D)
E)

Question 48

Use matrices to solve the system.

A)
B)
C)
D)The system is dependent
E)No solution

Question 49

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Question 50

A stationary company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $1.2, and a regular notebook, which sells for $0.85. The production cost is $1.00 for each deluxe notebook and $0.75 for each regular notebook. The company has the facilities to manufacture between 2,000 and 3,000 deluxe and between 3,000 and 6,000 regular notebooks, but not more than 8,000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production cost?

A)2,300 deluxe and 5,000 regular
B)3,000 deluxe and 6,000 regular
C)2,000 deluxe and 6,000 regular
D)3,000 deluxe and 5,000 regular
E)2,500 deluxe and 5,000 regular

Question 51

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 52

Three average monthly low temperatures for Detroit are listed in the table.
Let correspond to January, to February, . . . , and to December. Determine a quadratic function that interpolates the data - that is, determine the constants a, b, and c such that , and .

A)
B)
C)
D)
E)

Question 53

Use the method of substitution to solve the system.

A)( - 3, -16 )
B)( - 3, -16 ), ( 6, 11 )
C)( 6, 11 )
D)( - 3, -16 ), ( 5, 11 )
E)no solution

Question 54

Use the method of substitution to solve the system.

A)( 7, 0 ), ( 3, 2 )
B)( 7, 0 )
C)( 3, 2 )
D)( 7, 1 ), ( 3, 2 )
E)no solution

Question 55

Find the determinant of the matrix.

A)1,616.95
B)4,850.85
C)3,233.9
D)6,467.8
E)461.99

Question 56

Use the method of substitution to solve the system.

A)
B)
C)
D)
E)no solution

Question 57

A small firm manufactures bookshelves and desks for microcomputers. For each product it is necessary to use a table saw and a power router. To manufacture each bookshelf, the saw must be used for hour and the router for 1 hour. A desk requires the use of each machine for 2 hours. The profits are $20 per bookshelf and $50 per desk. If the saw can be used for 8 hours per day and the router for 12 hours per day, how many bookshelves and desks should be manufactured each day to maximize the profit?

A)4 bookshelves and 2 desks
B)4 bookshelves and 4 desks
C)2 bookshelves and 4 desks
D)0 bookshelves and 2 desks
E)4 bookshelves and 0 desks

Question 58

Find the determinant.

A)
B)
C)
D)
E)

Question 59

A manufacturer of cell phones makes a profit of $24 on a deluxe model and $30 on a standard model. The company wishes to produce at least 80 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not exceed 200 phones. How many of each type should be produced daily in order to maximize the profit?

A)120 standard and 80 deluxe
B)60 standard and 140 deluxe
C)30 standard and 80 deluxe
D)120 standard and 60 deluxe
E)160 standard and 40 deluxe

Question 60

Find a system of inequalities for the graph shown.

A)
B)
C)
D)
E)

Question 61

Find the maximum and minimum values of the objective function on the region in the figure.

A)The maximum is at
,
The minimum is
at
.
B)The maximum is at
,
The minimum is
at
.
C)The maximum is at
,
The minimum is
at
.
D)The maximum is at
,
The minimum is
at
.
E)The maximum is at
,
The minimum is
at
.

Question 62

A manufacturer of tennis rackets makes a profit of $16 on each oversized racket and $10 on each standard racket. To meet dealer demand, daily production of standard rackets should be between 25 and 80, and production of oversized rackets should be between 8 and 33. To maintain high quality, the total number of rackets produced should not exceed 80 per day. How many of each type should be manufactured daily to maximize the profit?

A)47 standard and 8 oversized
B)33 standard and 47 oversized
C)47 standard and 33 oversized
D)25 standard and 8 oversized
E)25 standard and 33 oversized

Question 63

Three average monthly low temperatures for Detroit are listed in the table.
Let correspond to January, to February, . . . , and to December. Determine a quadratic function that interpolates the data - that is, determine the constants a, b, and c such that , and .

A)
B)
C)
D)
E)

Question 64

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 65

A hospital dietician wishes to prepare a corn-squash vegetable dish that will provide at least grams of protein and cost no more than cents per serving. An ounce of creamed corn provides gram of protein and costs cents. An ounce of squash supplies gram of protein and costs cents. For taste, there must be at least ounces of corn and at least as much squash as corn. It is important to keep the total number of ounces in a serving as small as possible. Find the combination of corn and squash that will minimize the amount of ingredients used per serving.

A) ounces of corn and ounces of squash
B) ounces of corn and ounces of squash
C) ounces of corn and ounces of squash
D) ounces of corn and ounces of squash
E) ounces of corn and ounces of squash

Question 66

Find the determinant.

A)
B)
C)
D)
E)

Question 67

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist

Question 68

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 69

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent.
E)The equations are dependent.

Question 70

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 71

Let be the identity matrix of order 2, and let . Find the polynomial for the given matrix A. (In the study of matrices, is the characteristic polynomial of A, and the zeros of are the characteristic values (eigenvalues) of A.)

A)
B)
C)
D)
E)

Question 72

Sketch the graph of the inequality.

A)
B)
C)
D)
E)

Question 73

Use Cramer's rule, whenever possible, to solve the system.

A)
B)
C)
D)The equations are dependent.
E)The system is inconsistent.

Question 74

The data in the table are generated by the function . Approximate the unknown constants a and b to four decimal places. x
1 2 3 4
F ( x )
0)67654 0.44668 0.29492 0.19472

A)
B)
C)
D)
E)

Question 75

Use the method of substitution to solve the system.

A)( - 1, -12 ), ( 8, 68 )
B)( 9, 68 )
C)( - 1, -12 )
D)( - 1, -12 ), ( 9, 68 )
E)no solution

Question 76

Find the values of b such that the system has no solution.

A)
B)
C)
D)
E)no solution

Question 77

Use matrices to solve the system.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 78

Solve the system using the inverse method.

A)
B)
C)
D)The system is inconsistent
E)The equations are dependent

Question 79

Find the determinant of the matrix.

A)721.50
B)7,575.75
C)2,525.25
D)10,101
E)5,050.5

Question 80

Find the inverse of the matrix if it exists.

A)
B)
C)
D)
E)Does not exist