1 What you will learn We need to review several concepts from Algebra II: Solving a system of equations graphically Solving a system of equations algebraically.

1

What you will learn

We need to review several concepts from Algebra II:

Solving a system of equations graphically Solving a system of equations algebraically

Matrix operations Matrix transformations Solving a system of equations using matrices

Graphing linear inequalities

Objective: Brain Dump of Chapter 2

2

Some Vocabulary

A system of two linear equations in two variables, x and y consists of two equation of the following form:

Ax + By = C

Dx + Ey = F

A solution of a system of linear equations is an ordered pair (x, y) that satisfies each equation.


3

How Do You Solve These Things?

The first method we will use is to graph the two equations and see where they intersect.

Example: Solve the system

2x – 3y = 1

x + y = 3

x-10 -5 5 10

y

-10

-5

5

10


4

Number of Solutions

x-10 -5 5 10

y

-10

-5

5

10

x-10 -5 5 10

y

-10

-5

5

10

x-10 -5 5 10

y

-10

-5

5

10

Infinite Number One Solution No SolutionSolutions


5

You Try! Use substitution to find the solution to

the following:3x – y = 132x + 2y = -10


6

You Try! Solve using the linear combination

method: 2x + 3y = 5x – 5y = 9


7

Many or No Solutions

When you solve a system of equations and you get something that is impossible (e.g. 6 = 7) then the system has no solution.

When you solve a system of equation and you get something that is always true (0 = 0), then the system has an infinite number of solutions.


8

Matrices

236

123 2 rows

3 columns

entry


9

Adding and Subtracting Matrices

You can add or subtract matrices only if they have the same dimensions!!

To add or subtract you simply add or subtract the corresponding entries.

A. B. C.

3

0

1

7

4

3

16

72

04

38

5

1

43

02


10

Scalar MultiplicationIn matrix algebra, a real number is often called a scalar. To multiply a matrix by a scalar, you multiply each entry in the matrix by the scalar. This process is called scalar multiplication.

A. B.

74

023

62

86

54

54

30

216

2


11

Multiplying Matrices…This isn’t for the fearful

In order to multiply two matrices A and B, the number of columns in A must be equal to the number of columns in B.

Example:

The answer matrix will be…

Question? Could I multiply matrix B times matrix A?

26

32

16

42

32


12

More Fun With Matrices

A = B = C =

1. Find AB 2. Find BA

3. Find A(B + C)

31

12

24

02

23

11


13

Transformations with Matrices You can dilate, translate, reflect and

rotate figures using matrices.

Dilate a triangle with vertices at X(0,8), Y(5,9), and Z (-3,2) by a scale factor of 2.


14

Translating

Translate quadrilateral ABCD 2 units to the left and 4 units up. A(-1, 1) B(4, 0) C(4, -5),and D(-1, -3)


15

Reflecting Using Matrices Figures can be reflected over the x axis, y axis

or over the line y = x. Use a reflection matrix and matrix multiplication.

Square ABCD has vertices at (-1,2), (-4,1),

(-3,-2) and D(0,-1). Find the image of the square after a reflection over the y-axis:

1

0

2

3

1

4

2

1

10

01


16

List of Transformation Matrices Page 90 and 91 in your book.

Remember, you put the transformation matrix first and then multiply it by the matrix with the x and y coordinates of the figures.


17

How Do You Get This Thing?

Determinant of a 2 x 2 matrix:

dc

ba

dc

ba

= ad - cb


18

Finding the Determinant for a 3 x 3

h

e

b

g

d

a

ihg

fed

cba

ihg

fed

cba

det =(aei+bfg+cdh)–(gec+hfa+idb)


19

Example Find the inverse of A

=

24

13

ac

bd

cbadac

bd

AA

111


20

We Have ArrivedSolving a system of equations using matrices involves setting up three matrices. The first matrix is a matrix that includes the coefficients (numbers “attached” to x and y). The second matrix contains the variables and acts only as a placeholder. The third matrix contains the constants, or “answer” side of the equations.


21

Setting Up the Matrices Write the system of linear equations as a

matrix equation:-3x + 4y = 52x – y = -10


22

What Next Remember:

AX = B

If we multiply both sides of the equation by the inverse of matrix A we can “solve” the equation.

10

5

12

43

y

x


23

Another Example Use a matrix equation to solve the

following2x + 3y + z = -13x + 3y + z = 12x + 4y + z = -2


24

Homework Start working on the Chapter 2 Review packet. It

will be due at the end of class Friday.

You will have a chance to do some in-class work on Wednesday.

We will do corrections and go over tricky problems on Friday.

We will have a practice “quest” on Monday.

You will have a quest on this material next Wednesday.

1 What you will learn We need to review several concepts from Algebra II: Solving a system of equations graphically Solving a system of equations algebraically.

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