chapter 2: systems of linear equations and inequalities...

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Pre-Calculus Calendar

Chapter 1: Linear Relations and Functions

Chapter 2: Systems of Linear Equations and Inequalities

Monday Tuesday Wednesday Thursday Friday

1/29 1.1 p. 10: 19-25, 33, 35, 37-41, 43, 45, 47-50

1/30 1.2 p. 17: 10-20 even, 21-24, 31

1/31

1.1-1.2 Review

2/1

1.1-1.2 Quiz *All 1.1-1.2 homework

must be handed in BEFORE the quiz to

receive full credit

2.1 p. 71: 15-29 odd, 28, 32, 35, 38

2/2 2.2 p. 76: 9-15 odd, 18, 22

2/5

2.1-2.2 Review



receive full credit

2/6 (Sandtveit out) 2.3 p. 83: 15-19 odd, 25-45 odd, 50, 52

2/7 2.4 p. 93: 11-23 odd, 26, 30, 34

2/8

2.3-2.4 Review

2/9



receive full credit

2.5 p. 102: 15-39 odd, 45, 48, 51

2/12 2.6 p. 110: 15-19 odd, 26

2/13

2.5-2.6 Review

2/14



receive full credit

2.7 p. 116: 12-15

2/15

2.7 Worksheet

2/16

Start Chapter 2 Test Review

½ Day

2/19 Mid-Winter Break

2/20 Mid-Winter Break

2/21

Chapter 2 Test Review

*All Chapter 2 Homework must be handed in

BEFORE the end of class to qualify for a test retake

2/22

Chapter 2 Test

2/23

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Success Criteria

I can determine whether or not a relation is a function. Page(s) ____

I can use interval notation to write the domain of a function Page(s) ____

I can add, subtract, multiply, divide, and compose (composition) functions. Page(s) ____

I can solve systems of equations by graphing. Page(s) ____

I can solve systems of equations by substitution. Page(s) ____

I can solve systems of equations by elimination. Page(s) ____

I can solve a system of 3 variables. Page(s) ____

I can add and subtract matrices. Page(s) ____

I can multiply matrices. Page(s) ____

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I can use matrices to perform reflections over the x and y axes Page(s) ____

I can use matrices to rotations of 90, 180, and 270 degrees Page(s) ____

I can use the Matrix Transformation Theorem to generate transformation matrices. Page(s) ____

I find the determinant of 2x2 and 3x3 matrices Page(s) ____

I can find the inverse of 2x2 matrices by hand; 2x2 and 3x3 with a calculator Page(s) ____

I can use a matrix equation to solve a system of equation with any number of variables Page(s) ____

I can solve a system of linear inequalities Page(s) ____

I can use linear programming to maximize and minimize variables Page(s) ____

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Name: __________________________________

Pre-Calculus Notes: Chapter 1 – Relations, Functions, and Graphs

Section 1 – Relations and Functions

Relation: a set of ordered pairs

Domain: x-value, input, independent variable

Range: y-value, output, dependent variable

Function: a relation in which each x-value is paired with exactly one y-value

Example 1

State the relation of the wind chill data as a set of ordered pairs. Also state the domain and range

of the relation.

Windchill Factors at 20oF

Wind Speed (mph) Windchill

Temperature (oF)

5 19

10 3

15 -5

20 -10

25 -15

30 -18

Example 2

The domain of a relation is all consecutive integers between -2 and 2. The range y of the relation is

2 less than twice x, where x is a member of the domain. Write the relation as a table of values and

as an equation. Then graph the relation.

x y

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Example 3

State the domain and range of each relation. Then state whether the relation is a function.

a. b.

c. {(-2, 0), (3, 2), (4, 5)} d. {(-3, -2), (-2, -2), (0, -2), (0, 2)}

Vertical Line Test

If every vertical line drawn on the graph of a relation passes through no more than one point of the

graph, then the relation is a function.

Example 5

Determine whether the graph of each relation represents a function. Explain.

a. b.

Function Notation

__________________________________________________________________________________________________

Example 6

Evaluate each function for the given value.

a. f(-1) if f(x) = -x3 – 1 b. g(3) if g(x) = 23x

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c. g(m) if g(x) = 2x6 – 10x4 – x2 + 5 d. h(a – 2) if h(x) = 2x2 – x + 3

Interval Notation

Intervals whose graphs are segments:

Interval notation Set notation

Closed interval from a to b

Open interval from a to b

Half-open interval from a to b, including a

Intervals whose graphs are rays:

Closed infinite interval

Open infinite interval

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Example 7

State the domain of each function.

a. f(x) = 92 x b. 4

1

xxg

c. 1

x

xxh d.

4

42

x

xxk

Section 2 – Composition of Functions

Operations with Functions:

Sum: (f + g)(x) = ______________ Difference: (f – g)(x) = ______________

Product: xgf = ______________ Quotient: xg

f

= ______________

Composition of Functions: Denoted ___________________ The domain of the composition includes all

of the elements in x in the domain of g for which f(x) is in the domain of f.

Example 1

Given f(x) = 2x – 1 and g(x) = x2, find each function.

a. (f + g)(x) b. (f – g)(x)

c. xgf d. xg

f

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Example 2

For the Lotsa Coffee Shop, the revenue r(x) in dollars from selling x cups of coffee is

r(x) = 1.5x. The cost c(x) for making and selling the coffee is c(x) = 0.2x + 110.

a. Write the profit function. b. Find the profit on 100, 200, and

500 cups of coffee sold.

Example 3

Find xgf and xfg for f(x) = x2 – 1 and g(x) = 3x.

Example 4

a. State the domain of xgf for 4 xxf and 2

1

xxg .

b. State the domain of xgf for 6 xxf and 2

12

x

xg .

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Name: __________________________________

Pre-Calculus Notes: Chapter 2 – Systems of Linear Equations and Inequalities

Section 1 – Solving Systems of Equations in Two Variables

System of equations ________________________________________________________________

Solution to the system ________________________________________________________________

Consistent system ________________________________________________________________

Independent system ________________________________________________________________

Dependent system ________________________________________________________________

Inconsistent system ________________________________________________________________

Systems of equations can be solved using one of three different methods:

Graphing

Substitution

Elimination Example 1

Solve the system of equations by graphing.

y = 4x – 18

y = 54

3x

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Example 2

Use the substitution method to solve the system of equations.

y = 3x – 8

2x + y = 22

Example 3

Use the elimination method to solve the system of equations.

5x + 2y = 340

3x – 4y = 360

Example 4

Madison is thinking about leasing a car for two years. The dealership says that they will lease her

the car she has chosen for $326 per month with only $200 down. However, if she pays $1600

down, the lease payment drops to $226 per month. What is the break-even point when comparing

these lease options? Which 2-year lease should she choose if the down payment is not a problem?

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Section 2 – Solving Systems of Equations in Three Variables

Example 1

Solve the system of equations.

3y = -9z

4x + 2y – 2z = 0

-3x – y + 4z = -2

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Example 2

Solve the system of equations.

5x – 2y + z = 11

2x + y + 3z = 0

6x – 2y – 2z = 16

Example 3

In the 1998 WNBA season, Sheryl Swoopes made 83% of her 86 attempted free throws. She made

244 of her 1-point, 2-point, and 3-point attempts, resulting in 453 points. Find the number of 1-

point free throws, 2-point field goals, and 3-point field goals Swoopes made in the 1998 season.

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Section 3 – Modeling Real-World Data with Matrices

Matrix ____________________________________________________________________________

nm matrix ____________________________________________________________________________

Dimensions ____________________________________________________________________________

33 matrix 12 matrix 32 matrix

163

380

542

7

5.0

85

311

352

1

There are special names for certain matrices:

Row Matrix ________________________________________________________________________

Column Matrix ________________________________________________________________________

Square Matrix ________________________________________________________________________

nth order ________________________________________________________________________

Matrix

________________________________________________________________________

Equal Matrices ________________________________________________________________________

Example 1

During the summer, Ms. Robbins received several types of grains on her farm to feed her livestock.

Use a matrix to represent the data.

June – 15,000 bushels corn, 2000 bushels soybeans, 500 bushels oats

July – 13,500 bushels corn, 6500 bushels soybeans, 1000 bushels oats

August – 14,000 bushels corn, 5500 bushels soybeans, 1500 bushels oats

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Exploring Matrix Operations - For each operation, determine the rule.

Matrix Addition

1.)

129

114

87

51

42

63 2.)

4150

1522

1156

732

306

814

Rule: _____________________________________________________________________________

Zero Matrix ________________________________________________________________________

Additive Inverse ________________________________________________________________________

Matrix Subtraction

3.)

45

12

87

51

42

63 4.)

21512

146

1156

732

306

814

Rule: _____________________________________________________________________________

Scalar Multiplication

5.)

3933

24180

15321

1311

860

517

3 6.)

45

35

9

75

Rule: _____________________________________________________________________________

Matrix Multiplication

7.)

1258

1859

69

08

25

34 8.)

156

2432

25

34

69

08

9.)

24330

422121

245

633

35

07 10.)

29

21

4

1

83

45

11.)

161422

12416

244

032

210

513

248

Rule: _____________________________________________________________________________

_____________________________________________________________________________

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Example 2

Find the values of x and y for which the matrix equation

12

4

3 x

x

y

y is true.

Example 3

Find A + B if

13

05

47

A and

52

98

106

B .

Example 4

Find S – T if

824

312S and

487

145T .

Example 5

If

91

83

25

A , find 4A.

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Example 6

Use matrices

423

010

214

A ,

32

24B ,

013

321C to find each product.

a.) BC b.) CB

c.) AC d.) CA

Example 7

At Ohio State University, professional students pay different tuition rates based on the programs

they have chosen. For the 2002-03 school year, in-state students in the school of medicine paid

$5646 per quarter, dental school students paid $4792 per quarter, and veterinary medicine

students paid $4405 per quarter. The chart lists the total student enrollment in those programs for

each quarter of the 2002-03 school year. Use matrix multiplication to find the amount of tuition

paid for each of these four quarters.

Quarter Enrollment

Med. Dent. Vet.

Autumn 826 400 537

Winter 818 401 537

Spring 820 399 536

Summer 425 205 135

Source: The Ohio State University Registrar

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Section 4 – Modeling Motion with Matrices

Transformations

translations ________________________________

reflections ________________________________

rotations ________________________________

dilations ________________________________

Triangle ABC can be represented by the following vertex matrix.

Triangle A’B’C’ is congruent to and has the same orientation as ABC , but is moved

___________________________________________________ from ABC ’s location. The coordinates

of ''' CBA can by expressed as the following vertex matrix:

Compare the two matrices. If you add

to the first matrix you get the second matrix.

This type of matrix is called a ______________________________________. In this

transformation ABC is the ______________________ and ''' CBA is the _______________

after the translation.

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Example 1

Suppose the quadrilateral RSTU with vertices R(3, 2), S(7, 4), T(9, 8), and U(5, 6) is translated 2

units right and 3 units down.

a.) Represent the vertices of the quadrilateral as a matrix.

b.) Write the translation matrix.

c.) Use the translation matrix to find the vertices of R’S’T’U’, the translated image of the

quadrilateral.

d.) Graph the quadrilateral RSTU and its image.

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Reflections over the x-axis

Reflect the point (1,2) over the x-axis.

How did the coordinate change?

What matrix could you multiply

2

1

by to yield your new coordinate?

Reflections over the y-axis

Reflect the point (1,2) over the y-axis.



2

1


Reflections over the line y = x

Reflect the point (1,2) over the line y = x.



2

1


Reflection Matrices

For a reflection over the: Symbolized by: Multiply the vertex matrix by:

x-axis

y-axis

line y = x

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Rotations about the origin of 90o

Rotate the point (1,2) about the origin 900.



2

1






2

1






2

1


Rotation Matrices

For a counterclockwise rotation

about the origin of

Symbolized by: Multiply the vertex matrix by:

90o

180o

270o

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Matrix Transformation Theorem

Example 2

Use a reflection matrix to find the coordinates of a reflection over the y-axis of square SQAR with

vertices S(4, 1), Q(7, 3), A(9, 0), R(6, -2). Then graph the pre-image and the image on the same

coordinate grid.

Example 3

An animated figure rotates about the origin. The image has key points at (5, 2), (3, -1),

(2, -4), (-1, 2), and (2.5, 1.5). Find the locations of these points at the 90o, 180o, and 270o

counterclockwise rotations.

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Example 4

A parallelogram has vertices W(-2, 4), X(0, 8), Y(4, 6), and Z(2, 2). Find the coordinates of the

dilated parallelogram W’X’Y’Z’ for a scale factor of 1.5. Describe the dilation.

Section 5 – Determinants and Multiplicative Inverses of Matrices

Our goal is to solve a system of equations using matrices, but before we can do that, we need some

math magic.

Solve the system of equations by using matrix equations.

4x – 2y = 16

x + 6y = 17

Each square matrix has a _____________________________. The determinant of

67

48 is a number

denoted by 67

48 or det

67

48.

Second-Order

Determinant ______________________________________________________________________

Third-Order

Determinant ________________________________________________________________________

Example 1

Find the value of 68

20

.

23

Example 2

Find the value of

730

846

135

.

Identity Matrix ________________________________________________________________________

for Multiplication

________________________________________________________________________

Identity Matrix for

Second-Order Matrix __________

Inverse Matrix A-1 ________________________________________________________________________

________________________________________________________________________

Inverse of a

Second-Order

Matrix ________________________________________________________________________

Example 3

Find the inverse of the matrix

13

98.

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Example 4

Solve the system of equations by using matrix equations.

4x – 2y = 16

x + 6y = 17

Section 6 – Solving Systems of Linear Inequalities

Example 1

Belan Chu is a graphic artist who makes greeting cards. Her startup cost will be $1500 plus $0.40

per card. In order for her to remain competitive with large companies, she must sell her cards for

no more than $1.70 each. How many cards must Ms. Chu sell in order to make a profit?

Polygonal

Convex Set _______________________________________________________________________

_______________________________________________________________________

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Example 2

Solve the system of inequalities by graphing and name the coordinates of the polygonal convex set.

5

0

0

yx

y

x

Vertex Theorem _____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

Example 3

Find the maximum and minimum values of f(x, y) = y – 2x + 5 for the polygonal convex set

determined by the system of inequalities.

5

2

142

8

1

yx

y

yx

y

x

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Section 7 – Linear Programming

Linear 1. Define all variables.

Programming

Procedure 2. Write the constraints as a system of inequalities.

3. Graph the system and find the coordinates of the vertices of the

polygon formed.

4. Write an expression whose value is to be minimized or maximized.

5. Substitute values from the coordinates of the vertices into the

expression.

6. Select the greatest or least result.

Example 1

Suppose a lumber mill can turn out 600 units of product each week. To meet the needs of its

regular customers, the mill must produce 150 units of lumber and 225 units of plywood. If the

profit for each unit of lumber is $30 and the profit for each unit of plywood is $45, how many units

of each type of wood product should the mill produce to maximize profit?

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Example 2

The profit on each set of cassettes that is manufactured by MusicMan, Inc., is $8. The profit on a

single cassette is $2. Machines A and B are used to produce both types of cassettes. Each set

takes nine minutes on Machine A and three minutes on Machine B. Each single takes one minute

on Machine A and one minute on Machine B. If Machine A is run for 54 minutes and Machine B is

run for 42 minutes, determine the combination of cassettes that can be manufactured during the

time period that most effectively generates profit within the given constraints.

Example 3

The Woodell Carpentry Shop makes bookcases and cabinets. Each bookcase requires 15 hours of

woodworking and 9 hours of finishing. The cabinets require 10 hours of woodworking and 4.5

hours of finishing. The profit is $60 on each bookcase and $40 on each cabinet. There are 70

hours available each week for woodworking and 36 hours available for finishing. How many of each

item should be produced in order to maximize profit?

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Example 4

A manufacturer makes widgets and gadgets. At least 500 widgets and 700 gadgets are needed to

meet minimum daily demands. The machinery can produce no more than 1200 widgets and 1400

gadgets per day. The combined number of widgets and gadgets that the packaging department can

handle is 2300 per day. If the company sells both widgets and gadgets for $1.59 each, how many of

each item should be produced in order to maximize profit?

chapter 2: systems of linear equations and inequalities...

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