You solved systems of equations algebraically and represented data using matrices. (Lessons 0-5 and 0-6)
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1 You solved systems of equations algebraically and represented data using matrices. (Lessons 0-5 and 0-6) Solve systems of linear equations using matrices and Gaussian elimination. Solve systems of linear equations using matrices and Gauss-Jordan elimination.
2 multivariable linear system row-echelon form Gaussian elimination augmented matrix coefficient matrix reduced row-echelon form Gauss-Jordan elimination
3
4 Gaussian Elimination with a System Write the system of equations in triangular form using Gaussian elimination. Then solve the system. x + 3y + 2z = 5 3x + y 2z = 7 2x + 2y + 3z = 3
5 Gaussian Elimination with a System
6 Write the system of equations in triangular form using Gaussian elimination. Then solve the system. x 2y + z = 3 2x 3y + z = 3 x 2y + 2z = 5 A. x 2y + z = 3 y 2z = 3 z = 2; ( 1, 1, 2) C. x + 2y z = 3 y 2z = 5 z = 2; (7, 1, 2) B. x + 2y z = 3 y 2z = 5 z = 2; ( 1, 1, 2) D. x 2y + z = 3 y + 2z = 5 z = 2; ( 17, 9, 2)
7 Write an Augmented Matrix Write the augmented matrix for the system of linear equations. x + y z = 5 2w + 3x z = 2 2w x + y = 6
8 Write an Augmented Matrix
9 Write the augmented matrix for the system of linear equations. w + 2x + z = 2 x + y z = 1 2w 3x + y + 2z = 2 w + 3y z = 5 A. C. B. D.
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12 Identify an Augmented Matrix in Row-Echelon Form A. Determine whether is in row-echelon form.
13 Identify an Augmented Matrix in Row-Echelon Form B. Determine whether is in row-echelon form.
14 Identify an Augmented Matrix in Row-Echelon Form C. Determine whether is in row-echelon form.
15 Which matrix is not in row-echelon form? A. B. C. D. A. A B. B C. C D. D
16 Gaussian Elimination with a Matrix RESTAURANTS Three families ordered meals of hamburgers (HB), French fries (FF), and drinks (DR). The items they ordered and their total bills are shown below. Write and solve a system of equations to determine the cost of each item.
17 Gaussian Elimination with a Matrix
18 Gaussian Elimination with a Matrix
19 MOVIES A movie theater sells three kinds of tickets: senior citizen (SC), adult (A), and children (C). The table shows the number of each kind of ticket purchased by three different families and their total bills. Write and solve a system of equations to determine the cost of each kind of ticket. A. senior citizen: $5, adult: $8, children: $2 B. senior citizen: $5, adult: $7, children: $4 C. senior citizen: $6, adult: $8, children: $2 D. senior citizen: $4, adult: $7, children: $6
20 Use Gauss-Jordan Elimination Solve the system of equations. x y + z = 3 x + 2y z = 2 2x 3y + 3z = 8
21 Use Gauss-Jordan Elimination
22 Solve the system of equations. 2x y + z = 7 x + y z = 4 y + 2z = 4 A. x = 1, y = 2, z = 3 B. x = 1, y = 2, z = 3 C. x = 1, y = 2, z = 3 D. x = 1, y = 2, z = 3
23 No Solution and Infinitely Many Solutions A. Solve the system of equations. x + 2y + z = 8 2x + 3y z = 13 x + y 2z = 5
24 No Solution and Infinitely Many Solutions
25 No Solution and Infinitely Many Solutions B. Solve the system of equations. 2x + 3y z = 1 x + y 2z = 5 x + 2y + z = 8
26 No Solution and Infinitely Many Solutions
27 Solve the system of equations. x + 2y + z = 2 x + y 2z = 0 2y + 6z = 8 A. ( 2, 0, 4) B. no solution C. ( 3z, 2z, z) D. ( 5, 5, 3)
28 Infinitely Many Solutions Solve the system of equations. 4w + x + 2y 3z = 10 3w + 4x + 2y + 8z = 3 w + 3x + 4y + 11z = 11
29 Infinitely Many Solutions
30 Solve the system of equations. w x + y z = 2 2w x + 2y z = 7 w 2x y 2z = 1 A. (1, z 1, 2, z) B. ( 1, z + 1, 2, z) C. (z + 2, z + 1, z, z) D. no solution
31 You performed operations on matrices. (Lesson 0-5) Multiply matrices. Find determinants and inverses of 2 2 and 3 3 matrices.
32 identity matrix inverse matrix inverse invertible singular matrix determinant
33
34 Multiply Matrices A. Use matrices find AB, if possible. and to
35 Multiply Matrices B. Use matrices find BA, if possible. and to
36 Use matrices A = AB, if possible. A. B. C. D. and B = to find
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38 Multiply Matrices FOOTBALL The number of touchdowns (TD), field goals (FG), points after touchdown (PAT), and twopoint conversions (2EP) for the three top teams in the high school league for this season is shown in the table below. The other table shows the number of points each type of score is worth. Use the information to determine the team that scored the most points.
39 Multiply Matrices
40 CAR SALES A car dealership sells four types of vehicles; compact cars (CC), full size cars (FS), trucks (T), and sports utility vehicles (SUV). The number of each vehicle sold during one recent month is shown in the table below. The other table shows the selling price for each of the vehicles. Which vehicle brought in the greatest revenue during the month?
41 A. compact cars B. full size cars C. trucks D. sports utility vehicles
42
43 Solve a System of Linear Equations Write the system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. 2x1 + 2x2 + 3x3 = 3 x1 + 3x2 + 2x3 = 5 3x1 + x2 + x3 = 4
44 Solve a System of Linear Equations
45 Write the system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve the system. 2x1 x2 + x3 = 1 x1 + x2 x3 = 2 x1 2x2 + x3 = 2
46 A. ; ( 1, 2, 3) B. ; (1, 2, 3) C. ; ( 1, 2, 3) D. ; (1, 2, 3)
47
48 Verify an Inverse Matrix Determine whether inverse matrices. and are
49 Which matrix below is the inverse of A = A. A. A B. B C. C D. D B. C. D.?
50 Inverse of a Matrix A. Find A 1 when does not exist, write singular., if it exists. If A 1
51 Inverse of a Matrix B. Find A 1 when does not exist, write singular., if it exists. If A 1
52 Find A 1 when not exist, write singular. A. B. C. D., if it exists. If A 1 does
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55 Determinant and Inverse of a 2 2 Matrix A. Find the determinant of the inverse of the matrix, if it exists.. Then find
56 Determinant and Inverse of a 2 2 Matrix B. Find the determinant of the inverse of the matrix, if it exists.. Then find
57 Find the determinant of inverse, if it exists.. Then find its A. 2; C. 2; B. 2; D. 0; does not exist
58
59 Determinant and Inverse of a 3 3 Matrix Find the determinant of D 1, if it exists.. Then find
60 Find the determinant of A 1, if it exists.. Then find A. 3; C. 3, B. 3; D. 0; does not exist
61 You found determinants and inverses of 2 2 and 3 3 matrices. (Lesson 6-2) Solve systems of linear equations using inverse matrices. Solve systems of linear equations using Cramer s Rule.
62 square system Cramer s Rule
63
64 Solve a 2 2 System Using an Inverse Matrix A. Use an inverse matrix to solve the system of equations, if possible. 2x y = 1 2x + 3y = 13
65 Solve a 2 2 System Using an Inverse Matrix B. Use an inverse matrix to solve the system of equations, if possible. 2x + y = 9 x 3y + 2z = 12 5y 3z = 11
66 Use an inverse matrix to solve the system of equations, if possible. 2x 3y = 7 x y = 1 A. ( 2, 1) B. (2, 1) C. ( 2, 1) D. no solution
67 Solve a 3 3 System Using an Inverse Matrix COINS Marquis has 22 coins that are all nickels, dimes, and quarters. The value of the coins is $2.75. He has three fewer dimes than twice the number of quarters. How many of each type of coin does Marquis have?
68 MUSIC Manny has downloaded three types of music: country, jazz, and rap. He downloaded a total of 24 songs. Each country song costs $0.75 to download, each jazz song costs $1 to download, and each rap song costs $1.10 to download. In all he has spent $23.95 on his downloads. If Manny has downloaded two more jazz songs than country songs, how many of each kind of music has he downloaded? A. 6 country, 8 jazz, 10 rap B. 4 country, 6 jazz, 14 rap C. 5 country, 7 jazz, 12 rap D. 7 country, 9 jazz, 9 rap
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70 Use Cramer s Rule to Solve a 2 2 System Use Cramer s Rule to find the solution to the system of linear equations, if a unique solution exists. 4x 1 5x 2 = 49 3x 1 + 2x 2 = 28
71 Use Cramer s Rule to find the solution of the system of linear equations, if a unique solution exists. 6x + 2y = 28 x 5y = 14 A. no solution B. ( 4, 2) C. (4, 2) D. ( 4, 2)
72 Use Cramer s Rule to Solve a 3 3 System Use Cramer s Rule to find the solution of the system of linear equations, if a unique solution exists. y + 4z = 1 2x 2y + z = 18 x 4z = 7
73 Use Cramer s Rule to Solve a 3 3 System
74 Use Cramer s Rule to find the solution of the system of linear equations, if a unique solution exists. x y + 2z = 3 2x z = 3 3y + z = 10 A. (2, 3, 1) B. ( 2, 3, 1) C. (2, 3, 1) D. no solution
75 You graphed rational functions. (Lesson 2-4) Write partial fraction decompositions of rational expressions with linear factors in the denominator. Write partial fraction decompositions of rational expressions with prime quadratic factors in the denominator.
76 partial fraction partial fraction decomposition
77 Rational Expression with Distinct Linear Factors Find the partial fraction decomposition of.
78 Rational Expression with Distinct Linear Factors
79 Find the partial fraction decomposition of the rational expression. A. B. C. D.
80 Improper Rational Expression Find the partial fraction decomposition of.
81 Improper Rational Expression
82 Find the partial fraction decomposition of. A. B. C. D.
83 Denominator Find the partial fraction decomposition of
84 Denominator
85 Find the partial fraction decomposition of. A. B. C. D.
86 Denominator with Prime Quadratic Factors Find the partial fraction decomposition of
87 Denominator with Prime Quadratic Factors
88 Find the partial fraction decomposition of. A. B. C. D.
89
90 You solved systems of linear inequalities. (Lesson 0-4) Use linear programming to solve applications. Recognize situations in which there are no solutions or more than one solution of a linear programming application.
91 optimization linear programming objective function constraints feasible solutions multiple optimal solutions unbounded
92
93
94 Maximize and Minimize an Objective Function Find the maximum and minimum values of the objective function f(x, y) = 2x + y and for what values of x and y they occur, subject to the following constraints. x + y 5 2x + 3y 12 x 0 y 0
95 Maximize and Minimize an Objective Function
96 Find the maximum and minimum values of the objective function f(x, y) = x + 3y and for what values of x and y they occur, subject to the following constraints. 2x + 3y 12 2x + y 6 x 0 y 0 A. max of 12 at f(0, 4), min of 0 at f(0, 0) B. max of 3 at f(3, 0), min of 0 at f(0, 0) C. max of 10.5 at f(1.5, 3), min 3 of at f(3, 0) D. max of 12 at f(0, 4), min of 3 at f(3, 0)
97 Maximize Profit A. AUTOMOTIVE Mechanics at a repair garage carry two name brands of tires United, x, and Royale, y. The number of Royale tires sold is typically less than or equal to twice the number of United tires sold. The shop can store at most 500 tires at one time. Due to factory capacity, the number of Royale tires produced is greater than or equal to 50 more than 0.25 times the number of United tires. The garage earns $25 profit for each United tire and $20 profit for each Royale tire. Write an objective function and a list of constraints that model the given situation.
98 Maximize Profit
99 Maximize Profit B. AUTOMOTIVE Mechanics at a repair garage carry two name brands of tires United, x, and Royale, y. They typically sell at least twice the number of Royale tires as United. The shop can store at most 500 tires at one time. Due to factory capacity, the number of Royale tires produced is greater than or equal to 50 more than 0.25 times the number of United tires. The garage earns $25 profit for each United tire and $20 profit for each Royale tire. Sketch a graph of the region determined by the constraints to find how many of each tire the garage should sell to maintain optimal profit.
100 Maximize Profit
101 THEATER A local high school drama club is selling tickets to their spring play. A student ticket, x, costs $5 and a nonstudent ticket, y, costs $7. The auditorium has 522 seats. Based on current ticket sales the number of nonstudent tickets sold is less than or equal to half the number of student tickets sold. Write an objective function and a list of constraints that model the given situation. Determine how many of each kind of ticket the drama club needs to sell to maximize it s profit. What is the maximum profit?
102 A. B. C. D.
103 Optimization at Multiple Points Find the maximum value of the objective function f (x, y) = 2x + 2y and for what values of x and y it occurs, subject to the following constraints. y + x 7 x 5 y 4 x 0 y 0
104 Optimization at Multiple Points xxx-new art
105 Find the maximum value of the objective function f (x, y) = 3x + 3y and for what values of x and y it occurs, subject to the following constraints. y + x 9 x 6 y 4 x 0 y 0 A. f(x, y) = 18 at (6, 0) B. f(x, y) = 27 at (5, 4) and (6, 3) C. f(x, y) = 27 at (5, 4), (6, 3), and every point on the line y = x + 9 for 3 x 4 D. f(x, y) = 27 at (5, 4), (6, 3), and every point on the line y = x + 9 for 5 x 6.
106 Unbounded Feasible Region A. WAREHOUSE The employees of a warehouse work 8-hour shifts. There are two different shifts the employees can work, the day shift from 8 A.M. to 4 P.M. or the second shift from 2 P.M. to 10 P.M. Employees earn $11.50 per hour for the day shift and $13 for second shift. The day shift must have at least 35 employees. The second shift must have at least 25 employees. For the overlapping time, from 2 P.M. to 4 P.M., there must be at least 65 employees working. Write an objective function and list the constraints that model the given situation.
107 Unbounded Feasible Region
108 Unbounded Feasible Region B. WAREHOUSE The employees of a warehouse work 8-hour shifts. There are two different shifts the employees can work, the day shift from 8 A.M. to 4 P.M. or the second shift from 2 P.M. to 10 P.M. Employees earn $11.50 per hour for the day shift and $13 for second shift. The day shift must have at least 35 employees. The second shift must have at least 25 employees. For the overlapping time, from 2 P.M. to 4 P.M., there must be at least 65 employees working. Sketch a graph of the region determined by the constraints to find how many day-shift and second-shift employees should be scheduled to optimize labor costs.
109 Unbounded Feasible Region
110 YEARBOOK A high school s yearbook must contain at least 100 pages. At least 15 pages must contain color, x, and at least 30 pages must be black and white, y. Pages with color cost $9 each to format and pages that are black and white cost $8 dollars each to format. Write an objective function and a list of constraints that model the given situation. Determine how many of each kind of page is needed to minimize the cost of formatting the yearbook. What is the minimum cost?
111 A. f(x, y) = 8x + 9y; x + y 100, x 15, y 30; 15 pages with color, 85 black and white pages, $885 B. f(x, y) = 9x + 8y; x + y 100, x 15, y 30; 15 pages with color, 85 black and white pages; $815 C. f(x, y) = 9x + 8y; x + y 100, x 15, y 30; 15 pages with color, 85 black and white pages, $870 D. f(x, y) = 9x + 8y; x + y 100, x 15, y 30; 85 pages with color, 15 black and white pages, $885
6-1 Study Guide and Intervention Multivariable Linear Systems and Row Operations
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