0

Honors Algebra 2 ~ Spring 2014 Name_________________________

Unit 2: Systems of Equations Objectives: Students will be able to . . . 1. Solve systems of equations by graphing, substitution, elimination, and inverse matrix.

2. Use a graphing calculator to graph and solve systems of linear inequalities.

3. Find the maximum and minimum values of a function over a region using linear programming.

4. Solve a system of three equations in three variables.

5. Organize data into matrices; add, subtract, and multiply matrices.

6. Solve matrix equations.

7. Represent translations, rotations, dilations, and reflections with matrices.

8. Find and use inverse matrices and the identity matrix.

9. To solve problems using linear programming.

Day DATE LESSON ASSIGNMENT

1 Friday

Feb. 7

“What do you know about Solving Systems?”

Sections 3.1-3.3: Solving Systems of

equations Graphically and Algebraically,

Section 3.3: Graphing systems of inequalities

Packet p. 1

2 Monday

Feb. 10 Section 3.4: Linear Programming Packet p. 2

3 Tuesday

Feb. 11

Day 2 of Linear Programming

Packet p. 3

Study for Quiz

4 Wednesday

Feb. 12

Quiz on 3.1-3.4

“What do you know about Matrices”

Sections 4.1-4.3: Introduction to matrices;

Adding, subtracting, and scalar multiplication,

Packet p. 4

5 Thursday

Feb. 13

Section 4.3: Multiplication of matrices and

applications

CLASSWORK: Text pg.183-184:

# 22-24, 35, 38, 50, 55, 56

Determinants for a 2 x 2

Packet p. 5

Study for Quiz

6 Friday

Feb. 14

Section 4.5 Determinants for a 3 x 3

Section 4.6 Identity and Inverse Matrix

EARLY RELEASE Packet p. 6

7 Monday

Feb. 17

Quiz on Matrices

Section 4.7: Systems with 3 variables and

Applications(distance, chemistry, interest)

Pg. 7

8 Tuesday

Feb. 18

Warm-Up: Packet p. 7 Review Problem

Applications & Review for TEST

“3 equations, 3 unknowns”

Review Sheets Pg. 9 & 10

Study for TEST

9 Wednesday

Feb. 19

TEST

TBD

Print Unit 3 Notes & Packet!!

Objectives covered:

1.04 & 2.10

1

Honors Algebra 2 Unit 2: Homework Day 1

Solve by graphing. 1

1) 2 2) 5 11 3) 4 3 16 4) 32

1 10 1 4

y x x y x y y x

x y x y x y y 34

x

Solve by elimination or substitution. Show your work.

5. Einstein bought 8 oranges and one grapefruit for a total of $4.60. Later that day, he bought six

oranges and three grapefruits for a total of $4.80. Write a system of equations. Solve the system of

equations to find the price of an orange and grapefruit.

6. There are a total of 15 apartments in two buildings. The difference of two times the number of

apartments in the first building and three times the number of apartments in the second building is 5.

Write a system of equations to model the relationship between the number of apartments in the first

building and the number of apartments in the second building. How many apartments are in each building?

7. The perimeter of the square at the right is 72 units. What are the values of x and y?

8) 0.3 0.4 0.8 9) 1.2 1.4 11

0.7 0.8 6.8 0.4 0.3 0.9

x y x y

x y x y

Write a system of inequalities and solve by graphing.

10. Jacob is buying two kinds of notebooks for school. A spiral notebook costs $2, and a three-ring

notebook costs $5. Jacob needs at least six notebooks. The cost of the notebooks can be no more than

$20. Write a system, graph, and solve.

11. A camp counselor needs no more than 30 campers to sign up for two mountain hikes. The counselor

needs at least 10 campers on the low train and at least 5 campers on the high trail. Write a system of

inequalities to model the situation, graph, and solve.

10. 11.

2x

3y

2

Homework Day 2: Linear Programming

Graph each system of constraints. Name all vertices. Then find the values of x and y that

maximize or minimize the objective function. GRAPH on graph paper.

Find the values of x and y that maximize or minimize the objective function for each graph.

Then find the maximum or minimum value.

9. Open-ended: Write a system of constraints whose graphs determine a trapezoid. Write an

objective function and evaluate it at each vertex. Graph your constraints on graph paper.

REMEMBER: A trapezoid is a quadrilateral with atleast one pair of parallel lines.

3

Linear Programming Applications

1) You are about to take a test that contains questions of type A worth 4 points and of type B worth

7 points. You must answer at least 5 of type A and 3 of type B, but time restricts answering more

than 10 of either type. In total, you can answer no more than 18. How many of each type of

question must you answer, assuming all of your answers are correct, to maximize your score?

# of question A = ____________ # of question B = ___________________

2) Lois makes banana bread and nut bread to sell at a bazaar. A loaf of banana bread requires 2 c

flour and 2 eggs. A loaf of nut bread takes 3 c flour and 1 egg. Lois has 12 c flour and 8 eggs on

hand. She makes $2 profit per loaf of banana bread and $2 per loaf of nut bread. To maximize

profit, how many loaves of each type should she bake?

# of loaves of banana bread = _________ # of loaves of nut bread = _____________

3) Wheels Inc. makes mopeds and bicycles. Experience shows they must produce at least 10 mopeds.

The factory can produce at most 60 mopeds and 120 bicycles per month. The profit on a moped is

$134 and on a bicycle, $20. They can make at most 160 units combined. How many of each should

they make per month to maximize profit?

# of mopeds = _________ # of bicycles = ___________

4) Kay grows and sells tomatoes and green beans. It costs $1 to grow a bushel of tomatoes, and it

takes 1 yd2 of land. It costs $3 to grow a bushel of beans, and it takes 6 yd2 of land. Kay’s budget

is $15, and she has 24 yd2 of land available. If she makes $1 profit on each bushel of tomatoes and

$4 profit on each bushel of beans, how many bushels of each she grow to maximize profits?

# of bushels of tomatoes = _________ # of bushels of green beans = ________

5) A company makes whole wheat crackers and sesame crackers. The crackers are sold by the box.

Each box contains 5 packets of whole wheat crackers or 3 packets of sesame crackers. The

company cannot produce more than 150 packets of crackers per minute, but at least 15 boxes of

whole wheat crackers and at least 20 boxes of sesame crackers must be produced per minute. If

the profit per box of whole wheat crackers is 10 cents and the profit per box of sesame crackers

is 5 cents, how many boxes of each type should be produced per minute in order to maximize

profits?

# of boxes of whole wheat crackers = _________

# of boxes of sesame crackers = __________

4

2 3 6

4 2 12

x y

x y

6 3 5 4 3 1

2 5 1 0 5 5

3 7 4 3 2 8

4 2

1

6

x

y z

w v

4

5 3

0 8

a c

d

f

4 3 6 5 2 1

0 2 4 3 1 0

5 1 3 2 0 1

2 3 5

4 2 1

3 0 2

4 2 0

2 3 2

1 1 1

Intro to Matrices

1. For the matrix A =

6 4 0 2

1 3 5 1

2 6 1 0

3 5 2 7

5 2 4 1

a. State the dimensions of the matrix.

b. How many elements are in the matrix?

c. List the elements in the third row.

d. List the elements in the fourth column.

e. What element is in the third column

and the second row?

f. The element in the fifth row and first

column is _____ .

2. In the matrices given below, find the value of each variable which will make the matrices

equal. A = B =

3. Solve for x and y: 8. Add:

4. If A = 4 7 2

1 0 3 and B =

3 5 4

5 3 6 find A + B, A – B, B + A, B – A

5. Solve for the matrix X. 6. If A = and B =

X +

Calculate:

a. 2A + 3B b. A - -2B

Solve each equation for each variable.

7.

4 2 3 4 11 2 1 0

4 2 3 8 2 3

2 1 14 1 0 3 2 1

b d c

a

f g

8.

4 2 5 2 5 4

3 1 2 3

0 10 15 0 4 15

c d c d g

h f g

c

9. 2

2

9 44

2 52

x

yy

5

Matrix Applications 1. Matrix S gives the number of three types of cars sold in March by two car dealers, and matrix P

gives the profit for each type of car sold.

Dealer

1 2

S =

12 10

40 15

17 42

P = $400 $650 $900

(A) Which matrix is defined, SP or PS? Find this matrix and interpret its elements.

2. A Chicago Company wants to send some of its key personnel to a convention in London. In the

company’s Research and Development Division, five people plan to fly first class, three people plan

to fly business class, and two people plan to fly coach class. In the Sales Division, four people

plan to fly business class, and eight people coach class.

(A) Display this information in a 2 x 3 travel matrix T.

Round-trip prices for four different airlines are as follows: Airline A charges $1,280 for

first class, $922 for business class, and $676 for coach. Airline B charges $1,400 for

first class, $1,024 for business class, and $728 for coach. Airline C charges $1,320 for

first class, $905 for business class, and $654 for coach. Airline D charges $1,450 for

first class, $1,050 for business class, and $734 for coach.

(B) Display this information in a price matrix P that can be multiplied with matrix T to

give the travel costs for each company division per airline.

(C) Find the product.

(D) How much will it cost to fly the Sales Division on Airline D?

(E) Which airline will cost the Research and Development Division the least?

*****************************************************************************************

Part 2: Find the determinant for each matrix.

3 4 3 9 1 41) 2) 3)

1 1 3 2

1 2 3 2 1 2

4) 0 1 3 5) 1 0 5

4 2 1 0 4 1

x x

y y a b

2 4 3

6) 3 0 2

1 3 0

6

PRACTICE: Matrices and Determinants

Determine the value of the determinant of each matrix.

1. -5 2

-8 -7 2.

-2 3 1

0 4 -3

2 5 -1

3.

0 -4 0

2 -1 1

3 -2 5

4.

2 -4 1

3 0 9

-1 5 7

5. 3 -4

7 9 6.

2 7 -6

8 4 0

1 -1 3

Solve for the variable.

7. 3 -4

302 5x

8. 2 -1

-163 4m

9.

3 -1

2 1 -2 10

4 1

x

x

10.

2 0 3

7 5 -1 -3 9

4 2 1

x

x

11. What is the value of 4 3

3 2

p q

p q when p = -2 and q = -1?

Find the inverse of each matrix. 2 2 4 7 3 4

12) 13) 14) 1 3 3 5 3 4

Determine whether the matrices are multiplicative inverses.

32 11

2 1 3 1 4 9 1 2215) , 16) , 17) , 3 1

1 25 3 5 2 2 6 3 42 2

3 3

7

REVIEW PROBLEM: Do on graph paper!!

The zoo is erecting a new habitat for the giraffes. A model of the habitat can

be drawn using the equations: y = 2x + 5, x = 2, y = 7, and y = x - 2. On graph paper,

prepare a drawing of the habitat. Identify the points on the graph which represent

the "corners" of the habitat. Find the AREA of the habitat. THINK!!

*********************************************************************************** Word Problems with 2 Variables:

1. Maria worked at a store selling CDs. The store she works in sells CDs for $10 and some for $12. She

knew that she sold 500 CDs at the end of the day, and there was $5750 in the cash register. How many

of each CD did she sell?

2. Barrett’s bookstore sells pencils for $0.10 each and erasers for $0.15 each. Last Tuesday, the store

sold 17 more pencils than erasers for a total of $23.45. How many of each item was sold?

3. Kimberly has 37 coins. She has only dimes and quarters, and the sum of her money is $5.65. How

many of each coin does she have?

4. A boat travels 36 miles downstream in 2 hours. The return trip takes 3 hours. Find the rate of the

boat in still water. Find the rate of the current.

5. A boat rowed for 10 miles downstream in 2 hours, and then rowed the same distance upstream 3 and

1/3 hours. Find the rate of the boat in still water. Find the rate of the current.

6. Part of an investment of $32,000 earns 7.5% annual interest; the rest 9%. If the annual interest

from both is $2670, how much is invested at the higher rate?

7. Sophie is buying party favors for her birthday party. The candles cost $1 each, the frames are $2

each, and the mugs are $2.50 each. She has $120 to spend on 75 favors. Also, she wants to buy twice

as many candles as mugs. How many frames should she buy?

8. With a tail wind an airplane can travel 1080 miles in 6 hours. Flying in the opposite direction with the

same wind blowing, the plane can fly 1/3 of that distance in ½ the time. Find the plane’s speed and wind

speed.

9. Jeff wants to fill nine 1-lb tins with a snack mix. He has $15 and plans to buy almonds for $2.45 per

lb, hazelnuts for $1.85 per lb, and raisins for $.80 per lb. Jeff wants the mix to contain an equal amount

of almonds and hazelnuts and twice as much of the nuts as the raisins by weight. Write a system of

equations and solve for how many of each ingredient Jeff should buy?

8

Matrices: Systems of Equations 1. Find two integers whose sum is 188 and difference is 34.

2. Two numbers differ by 45. Two-thirds of the larger number is 2 less than twice the smaller number. What are

the numbers?

3. Charlotte’s coin box contains only pennies and nickels. If she has 66 coins worth $2.38, how many of each type

of coin does she have?

4. Tickets to an all-star baseball game cost $3.00 for children under 12 and $4.50 for everyone else. If 225

tickets are sold for a total of $937.50, how many children’s tickets were sold?

5. A jeweler wants to make 1600 grams of 25% silver compound by mixing 20% and 40% silver compounds together.

How many grams of each kind will she need?

6. Two angles are complementary. The measure of the smaller angle is seven-eighths the measure of the larger

angle. Find the measure of the smaller angle.

7. The measure of the larger of two supplementary angles is 20° more than 7 times the measure of the smaller

angle. What is the measure of each angle?

8. Gilbert plans to invest$12,000 into two types of bonds which yield 9% and 11% annually. If he wants to earn a

total of $1200 annually, how much should be invest in each bond?

9. Enez recently invested $5000, part at 6.5% annual interest and the rest at 7%. How much was invested at each

rate if her annual income from both investments was $340?

10. In a small row boat, it took Alice 6 hours to go 6 miles upstream, but she was able to return to her starting

point in only 45 minutes. Assuming her rowing speed was constant, what was the speed of the current?

11. A stadium has 49,000 seats. Seats cost $25 in Section A, $20 in Section B, and $15 in Section c. The number

of seats in Section A equals the total of Sections B and C. Suppose the stadium takes in $1,052,000 from each

sold-out event. How many seats does each section hold?

12. You are designing a plumbing system for a new office building. Three pipes, A, B, and C enter the building from

the main water line. The total flow in all three pipes is 100 gal/min. If pipes B and C together carry 40 gal/min,

and pipe A carries twice as much water as pipe B, how much water must flow in each pipe?

9

Unit 2 Review

Honors Algebra 2

1. Solve by graphing: 2

0

x y

x y

2. Solve graphically: 4 16

4

x y

x y

3. Solve 10 5 2

210 6

y

x z

4. Graph the following system:

3

2 2

x

y x

5. Which point does not belong to the solution

set of the given system?

2 6

4 0

x y

x y a. (0, 6) b. (-3, 4)

c. (-4, 5) d. (-10, 1)

6. Solve 2 1 7

33 2 3

x y

y x

7. Solve the following system:

2 4 5 9

3 2 2

5 4 3 12

x y z

x y z

x y z

8. Solve:

2

3 3

6

x y z

x

x y

9. Buddy the bird is able to fly 48 miles with the

wind in 4 hours and 30 miles against the wind in 3

hours. Find Buddy’s speed and the rate of the wind.

10. Jack has $3.05 in quarters and dimes. He has 14

coins all together. How many quarters and how

many dimes does he have?

11. Solve: 2 3 6

4 6 3

x y

x y

12. Solve: 2 4 6

4 8 12

x y

x y

13. Solve: 4 30

2 12

x y

x y

14. Don has $12,000 to invest in AAA and bonds.

AAA bonds pay 6%. B bonds pay 9%. He wants to

invest at least twice as much in AAA bonds as in B

bonds. How much shall he invest in each type to

maximize his return? What is his return?

15. If the system 3 5 2

15 6

x y

y x is written as a matrix

equation, by which matrix would you multiply both

sides to obtain the solution?

A.

3 1

8 8

1 3

40 40

B.

3 1

8 8

1 3

16 16

C.

1 5

27 27

2 1

9 9

D.

1 5

33 33

2 1

11 11

16. If A2x4 B4x1 what are the dimensions of the

product?

17. If A2x6 B = C2x5 what are the dimensions of B? 21. __________

18. If D E7x1= G1x1 what are the dimensions of D?

19. Solve for matrix X if

7 9 0 7 9 0

17 1 14 17 1 14

8 4 2 6 8 3

X

10

0 0 2 6

1 3 2 11

1 2 1 8

X

20. Find the inverse of the matrix if it exists:

21. Evaluate each determinant of the following

matrices:

13 21 112 6

. . 2 4 14 3

17 2 0

a b

22. Is the multiplication of matrices commutative?

Show examples to prove, disprove, or both.

23. Solve each equation for each variable:

2 5 4 25 4

3 3 12 3 18

x

y y

24. Solve for X:

25. Write the system 2 3 18

6

x y

x y as a matrix

equation. Then find the solution.

26. Find the value of x for which 3 2 10

4 2 4

x y

y x

27. Evaluate 2 9

5 8

28. A florist creates 3 special floral arrangements.

One uses 3 lilies. The second uses 3 lilies and 4

carnations. The third uses 4 daisies and 3

carnations. Lilies cost $2.15 each, carnations cost

$0.90 each, and daisies cost $1.30 each.

A) Write a matrix to represent the number of each

type of flower in each arrangement.

B) Write a matrix to represent the cost of each type

of flower.

C) Find the matrix representing the cost of each

floral arrangement.

29. How many pounds of chocolate worth $1.20 a

pound must be mixed with chocolate worth 90 cents a

pound to produce a mixture of 10 pounds worth $1.00

a pound?

30. A coast-to-coast airplane trip takes 5 hours

heading East, with the wind, and 6 hours heading

West, against the wind. If the trip is 3000 miles each

way, find the speed of

the wind and the speed of the plane in still air.

31) Students sold 640 tickets to the school play.

Adult tickets cost $4 each and children’s tickets cost

$2 each. How many each of adult and student tickets

was sold if the ticket sales total $2084?

32. Fred plans to put a fence around a rectangular

lot. The length of the lot must be at least 52 feet.

The cost of the fence along the length of the lot is

$2 per foot, and the cost of the fence along the

width is $3 per foot. The total cost cannot exceed

$360.

a. Write a system of inequalities that models the

problem.

b. Graph the system and shade the feasible region.

c. What is the maximum width of the lot if the

length is 60 feet?

3 7

9 4

11

Review ANSWER KEY

1) (1, 1)

2)

3) x = -12, y = 4, z = -5

4)

5) A

6) (1, -2)

7) { (1, 2, 3)}

8) { (1, -7, 8)}

9) Buddy 11mph, Wind 1mph

10) 11 quarters and 3 dimes

11) No Solution Parallel lines

12) No Solution Parallel lines

13) (7, -2)

14) $4000 Bonds & $8000 AAA

15) D

16) 2 x 1

17) 6 x 5

18) 1 x 7

19)

0 0 0

34 0 28

14 12 5

20) 4 7

51 51

3 117 51

21) A) 18 B) -1087

22) Not always!

23) x = 15 y = 3

24)

1

2

3

25)

2 3 18

1 1 6

0

6

X

X

26) x = 4

27) 61

28) A) 1 3 0 0

2 3 4 0

3 0 3 4

lilies carnations daisies

Arrang

Arrang

Arrang

B) 2.15

0.90

1.30

Cost

lilies

carnations

daisies

C) 1 6.45

2 10.05

3 7.9

Cost

Arrang

Arrang

Arrang

29) 133

pounds of $1.20/pound

263

pounds of $0.9/pound

30) Plane 550mph Wind 50mph

31) 402 adult & 238 children

32.

Let x = length, y = width

2(2 ) 3(2 ) 360

52

0

Maximum width = 20 feet

x y

x

y