3-6 Reteaching Solving Systems Using Matrices
Chapter 3 Chapter Test (continued) Form G
2 4 23
1 22 27
3 5 0
† 23
5 3 S
Solve each system of equations.
13. c
5x 1 4y 2 z 5 1 2x 2 2y 1 z 5 1 2x 2 y 1 z 5 2
14. c
x 1 y 1 z 5 0 2x 1 3y 1 2z 5 21 x 2 y 1 z 5 2
15. c
x 1 2y 5 0 4x 2 z 5 4 5y 1 z 5 21
Do you unDerstanD?
16. List three methods used to solve systems of equations. Describe the strengths of each method.
17. You burn 4 Cal/min walking and 10 Cal/min running. You walk 10 to 20 min each day and run 30 to 45 min each day. You never spend more than an hour running and walking together. How much time should you spend on each activity to maximize the number of Calories you burn? Will you have exercised enough to burn off a 500 Calorie meal?
18. Plumber A charges $25 for a house call and $50 for each hour spent on the job.
Plumber B charges $35 for a house call and $45 for each hour spent on the job.
If your job will take 4 hours to complete, which plumber should you use? How much will it cost you?
19. Open Ended Write a system of inequalities that has infinite solutions.
20. Error Analysis A student says that the system of equations is represented by the matrix. What error did the student make? What is the correct matrix?
c
5x 2 2y 1 2z 5 7 3x 1 4y 5 11 2x 2 6y 1 5z 5 5
C
5 22 2
3 4 1
2 26 5 †
7 11 5 S
Chapter 3 Chapter Test
(continued) Form G(6, 23, 1)
The student used a 1 for the z-coefficient in the second equation instead of a 0.
Plumber B; $215
Answers may vary. Sample: 2x 2 3y K 7, 3y 2 2x L 27 yes, you will have burned off 510 Calories.
45 min running, 15 min walking;
(0, 1, 3) No unique solution
Graphing, substitution, elimination; Answers may vary. Sample: Graphing (2, 21, 4)
C
5 22 2
3 4 0
2 26 5 †
7 11 5 S allows you to visualize the solution. Substitution enables you to reduce the number of variables quickly. Elimination is applicable to any system and efficient with two variables.
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65
Chapter 3 Quiz 1
Form KLessons 3-1 through 3-3
Do you know HOW?
Solve each system by graphing.
1. ey 5 3x 1 4
y 5 3x 2 1 2. e6x 1 3y 5 29
22x 1 3y 5 15 3. e4x 1 y 5 4 y 2 x 5 4
Solve each system by substitution or elimination.
4. e2x 2 y 5 28
y 5 24x 1 2 5. e2x 2 y 5 5
3x 1 y 5 5 6. e3x 2 2y 5 2 3x 1 4y 5 50
Do you unDerstanD?
7. The community theater is selling tickets to its play. An adult ticket costs $12 and a child ticket costs $8. The theater wants to take in at least $2720 from ticket sales and has only 275 seats.
a. Write a system of inequalities to model the situation.
b. What is one possible combination of ticket sales that would satisfy the theater’s goal?
8. Reasoning Is it possible for a dependent linear system to consist of two lines with different slopes?
O y
x
22 4
4
4 2 4 O
y x
22
4 2
4
6
O y
x
2 4 6
2
2 4 no solution
(21, 6)
(23, 3)
(2, 21)
(0, 4)
(6, 8)
Variables may vary. Sample: ex 1 y K 275 12x 1 8y L 2720 Answers may vary. Sample: 180 adult tickets and 70 child tickets
No; a dependent system has two lines whose graphs are the same. If the lines have different slopes, then their graphs are not the same.
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66
Chapter 3 Quiz 2
Form KLessons 3-4 through 3-6
Do you know HOW?
Find the values of x and y that maximize or minimize the objective function for each graph.
1. •
y # 24x 1 9 2y # 2x 1 11 x $ 0, y $ 0
2. •
y # 22x 1 5 2y # 28x 1 16 x $ 0, y $ 0
3. •
y # 213x 1 7 y # 22x 1 12 x $ 0, y $ 0
Maximum for Minimum for Maximum for
P 5 2x 2 5y P 5 x 1 3y P 5 22x 1 7y
Solve each system by substitution or elimination.
4. •
5x 1 3y 2 z 5 21 22x 2 y 1 3z 5 4 4x 1 2y 1 z 5 21
5. •
2x 2 2y 1 4z 5 28 3x 1 y 2 4z 5 16 2x 2 3y 1 z 5 7
6. •
2x 2 y 1 3z 5 26 x 1 y 2 5z 5 13 24x 1 3y 2 z 5 4
Do you unDerstanD?
7. Error Analysis To represent the system •
2x 2 y 1 4z 5 11 x 1 2y 2 6z 5 211 3x 2 10z 5 5
, you picked Matrix A.
Your friend picked Matrix B. Which of you is correct? What mistake was made?
Matrix A Matrix B
C
2 21 4
1 2 26
3 0 210
† 11 211 5
S C
2 21 4
1 2 26
3 1 210
† 11 211 5 S
8. The sum of three numbers is 21. The second number is two more than twice the first number. The second number is three times the third number. What are the numbers?
(2.25, 0)
(23, 5, 1)
(0, 0)
(4, 0, 21)
(0, 7)
(1, 2, 22)
You are correct; your friend used 1 as the coefficient of a missing variable instead of 0.
first number: 5, second number: 12, third number: 4
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67
Chapter 3 Test
Form KDo you know HOW?
Solve each system by substitution or elimination.
1. e2x 2 y 5 7
6x 2 3y 5 14 2. e5x 1 2y 5 12
26x 2 2y 5 214 3. e5x 1 2y 5 28 4x 1 3y 5 2
Graph the solution of each system.
4. ey $ 22x 1 3
y , x 5. ey 2 1 # 3x
y 1 1 # x 6. ey . 2x 1 4 y # 2x 1 1
7. You have 13 bills in your wallet in $1, $5, and $10 bills. There are twice as many $1 bills as $5 bills. The number of $10 bills is one more than the number of $5 bills. How many of each bill do you have? How much money do
you have?
Graph the system of constraints. Identify all vertices. Find the values of x and y that maximize or minimize the objective function. Then find the maximum or minimum value.
8. •
y # 2x 1 7 4y # x 1 8 x $ 0, y $ 0
Maximize for P 5 5x 1 2y
y x
22 2
4
4 4
y
x
2 2
4
4 2 4
y
x
2 2 4 6
4 2
y
x 4
6
A(0, 0) 4 6 B(0, 2)
D(7, 0) C(4, 3)
2
no solution (2, 1) (24, 6)
six $1 bills, three $5 bills, four $10 bills; $61
max P at (7, 0) 5 35
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68
Chapter 3 Test
(continued) Form K9. What is the solution of the system represented by the matrix? C
2 21 2
3 2 21
21 23 2
† 12
0 11 S (24, 1, 3) (1, 24, 3) (3, 24, 1) (1, 3, 24)
Do you unDerstanD?
10. Writing Explain how you determine whether a system of linear equations is independent, dependent, or inconsistent without graphing the lines.
11. Mechanic A charges $45 for car repairs and $80 for each hour spent on your car.
Mechanic B charges $60 for repairs and $60 for each hour spent on your car.
a. If your car takes 5 hours to repair, which mechanic charges the least money?
b. How much will it cost you to have the work done by the less expensive mechanic?
12. At a bookstore, you spend $76 on 11 books and magazines. Books cost $8 each and magazines cost $5 each. Write a matrix that represents this system. How many books and how many magazines did you buy?
13. Reasoning The sum of three numbers is 15. The second number is twice the third number. Do you have enough information to determine the three numbers? If so, what are the three numbers? If not, what information do you still need? No; you need a third equation that defines another relationship between two or three of the numbers.
C
Rewrite both equations in slope-intercept form. If the lines have the same slope and same y-intercept, then they are equations of the same line, and the system is dependent. If the lines have the same slope but different y-intercepts, they are parallel lines, and the system is inconsistent. If the lines have different slopes, then the system is independent.
Mechanic B
$360
B1 1 8 5 ` 11
76R ; 7 books, 4 magazines
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69 task 1
Make three systems of two equations/inequalities from any in the box.
y 5 2x 1 1 y 5 5x 2 11 y $ 22x 1 4
2x 1 2y 5 2 y 5 213x 1 3 y 5 3x 1 1
y # 2x 1 4 y 5 2x 2 1 x 1 y 5 1
y 5 26x y $ 0
Use each method to solve one system. Show your work.
a. graphing b. substitution c. elimination
Then find a system that has each of the following.
d. coincident lines e. intersecting lines f. parallel lines g. perpendicular lines Explain your reasoning. Your models should present situations in which you make
comparisons and draw conclusions.