obj. 24 systems of linear equations (presentation)

8/2/2019 Obj. 24 Systems of Linear Equations (Presentation)

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Obj. 24 Systems of Linear

Equations

Unit 6 Systems of Equations and Inequalities


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Concepts and Objectives

Systems and Matrices (Obj. #24)

Set up an augmented matrix from a system ofequations

Solve an augmented matrix by calculator

Calculate the determinant of a square matrix Use Cramers Rule to solve a system of equations


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Systems of Linear Equations

A set of equations is called asystem of equations. If all of

the variables in all of the equations are of degree one,then the system is a linearsystem. There are three

possibilities:

There is a single solution that satifies all the

equations.

There is no single solution that satisfies all the

equations.

There are infinitely many solutions to the equations.


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Systems of Linear Equations

There are four different methods of solving a linear

system of equations:1. Substitution Solve one equation for one variable,

and substitute it into the other equation(s).

2. Elimination Transform the equations such that if

you add them together, one of the variables is

eliminated.

3. Graphing Graph the equations, and the solution is

their intersection.4. Matrices Convert the system into one or two

matrices and solve.


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Systems and Matrices

A matrixis a rectangular array of numbers enclosed in

brackets. Each number is called an elementof thematrix.

There are three different ways of using matrices to solve

a system:

Use the multiplicative inverse.

The Gauss-Jordan Method, which uses augmented

matrices.

Cramers Rule, which uses determinants.


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Augmented Matrices

In an augmentedmatrix, the coefficients and constants of

equations in standard form are combined into onematrix.

is written as

While we can solve this system manually, the calculator

makes this much easier, using a process called Reduced

Row-Echelon Form.

+ + =

+ =

+ + =

3 2 1

2 2

2

x y z

x y z

x y z

1 3 2 1

2 1 1 2

1 1 1 2


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Augmented Matrices

Example: Solve, using an augmented matrix.

=+ =

3 4 1

5 2 19

x y

x y


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Augmented Matrices

Solving augmented matrices on the NSpire is

ridiculously easy:1. Press theb key.

2. Select Matrix & Vector

3. Select Reduced Row-Echelon Form4. Press thet key.

5. Select the nn matrix

6. Change the rows to 2

7. Enter the matrix values and press.

Note: you can also just type in rref( and skip 1-3


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Augmented Matrices

Example: Write the system of equations associated with

the augmented matrix. Do not solve.

1 3 6 7

2 1 1 1

1 2 2 1

+ =

+ = + + =

3 6 7

2 1

2 2 1

x y z

x y z

x y z


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Determinants

Every n n matrixA is associated with a real number

called the determinantofA, written |A |. The determinant is the sum of the diagonals in one

direction minus the sum of the diagonals in the other

direction.

Example:

3 4

6 8= = 24 24 48( )( ) ( )( )= 3 8 6 4

a b

c dad cb=


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Determinants

Example: Find the determinant of

2 2

3 1

( )( ) ( )( )

=

2 2

2 1 3 23 1

= + =2 6 8


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Determinants

Example: Solve forx:

=3

4x

x x

=

2

3 4x x =

23 4 0x x

( )( ) + =4 1 0x x

=

4, 1x


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Determinants

To calculate the determinant of a 33 matrix, repeat the

first two columns to help you draw the diagonals:

Again, your calculator can also calculate the determinant

of a matrix you have entered. (Look for det on the

TI-83/84.)

8 2 4

7 0 3

5 1 2

= 7 0

5

8 2

1

4

3

8 2

7

2 5

0

1

= 500= ( )30+ 28+ (0 ( )24+ ( ))28+


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Cramers Rule To solve a system using Cramers Rule, set up a matrix of

the coefficients and calculate the determinant (D). Then, replace the first column of the matrix with the

constants and calculate that determinant (Dx).

Continue, replacing the column of the variable with the

constants and calculating the determinant (Dy, etc.)

The value of the variable is the ratio of the variable

determinant to the original determinant.


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Cramers Rule Example: Solve the system using Cramers Rule.

5

6 1

7 1

8

x y

x y

+ =

+ =

40 47

6 8

2 25

D = = =

71

18 7 15

8x

D = = =

( )15

65 6 11

1yD = = =

= = =

157.5

2

xDxD

= = =

11 5.52

yDyD


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Homework College Algebra & Trigonometry

Page 862: 5-14 HW: 5-14

Page 874: 8-36 (4s), 38, 62, 66, 74

HW: 8, 16, 20, 38, 62 Classwork: Algebra & Trigonometry

Page 141: 1-8, 13 (write out augmented matrix)

obj. 24 systems of linear equations (presentation)

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