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SYSTEM OF EQUATIONSSYSTEM OF EQUATIONS

& INEQUALITIES& INEQUALITIES

VIVIANA MARCELA BAYONAVIVIANA MARCELA BAYONA

CARDENASCARDENAS


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CONTENT

6.1 System of Linear Equations

6.11 Solve using inverse matrix

6.12 Solve using Cramers Rule 6.13 Solve using Gauss & Gauss Jordan

Elimination Method

6.2 System of Nonlinear Equations

6.3 System of Inequalities


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6.1 System of Linear Equations

By the end of this topic, you should be able to

Discuss system of linear equations and the types of

solution namely: unique, inconsistent and infinite

solutions.

Write a system of linear equations in matrix form

Solve a system of linear equation by using inverse

matrix, Cramers Rule, and Gauss & Gauss-Jordan

Elimination Method.


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What is system?

is an assemblage of

entity/objects, real or

abstract, comprising a

whole with each and

every component/element interacting or

related to another one.

Solar system, blood

system, computersystem, ext..


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System of Linear Equations

11 12 1 1 1

2 2 221 22

1 2

or

n

n

n mm m mn

a a a x b

a x ba a

x ba a a

!

- - -

AX = b

K

L

M MM M M

L

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

!

!

!

K

K

M

K

11 12 1

221 22

1 2

1 1

2 2

,

and

n

n

m m mn

n m

a a a

aa a

a a a

x b

x b

x b

! -

! - -

A

X = b

K

L

M M M

L

M M

The system of linear equations

Can be written in matrix form as

where


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Augmented Matrix

? A

11 12 1 1

2 221 22

1 2

or |

n

n

n

a a a b

a ba a

ba a a

-

A b

K

L

MM M M

L

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

!

!

!

K

K

M

K

11 12 1

221 22

1 2

1

2

,

and

n

n

m m mn

m

a a a

aa a

a a a

b

b

b

! -

! -

A

b

K

L

M M M

L

M

For the system of linear equations

The augmented matrix is given by,

where


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Types of solution

Linear systems

Homogenous AX = 0 AX = b

m = n .

. m < n

m > n

unique infinite

unique

m = n .

.

.

unique

m < n

m > n

infinite

unique

infinite

.

infinite

infinite None

None

Noneinfinite

0{A 0{A

0!A0!A

m n{ m n{

m Number of Row n Number of Column

Unique only 1 solution (the system is consistent)

Infinite many solution (the system is consistent)

None No solution (the system is not consistent)


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6.11 Solve using Inverse Matrix

Only for Square matrix

The formula given by:

1

1 1

1

1

From P re-multiply by

!

!

AX b A

A AX A b

IX A b

X A b

and 0m n! {A


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Examples 1 (Solve using Inverse Matrix)

1 2

1 2

3 2 6

5 4 8

x x

x x

!

!

1 2 3

1 2 3

1 2

2 2 1

3 2

2 3

x x x

x x x

x x

!

!

!

1 2

1 2

2 4

4 3 3

x x

x x

!

!

1 2

1 2 3

1 2 3

1

2 2 5

2 2 3

x x

x x x

x x x

!

!

!

1 2

3 4

Solve each of the following system of equality by Inverse Matrix


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6.12 Solve Using Cramers Rule

Only for Square matrix

The formula given by:

1 12 1 11 1 1

2 22 22 21 2

1 2

2 1

for 1, 2,...,

where , and so on

i

i

n n

n n

m m mn m m mn

x i n

b a a a b aa ab a a b

b a a a b a

! !

! ! - -

A

A

A A

K K

L L

M M M MM M

L L

and 0m n! {A


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Examples 2 (Solve Using Cramers Rule)

1 2

1 2

3 2 6

5 4 8

x x

x x

!

!

1 2 3

1 2 3

1 2

2 2 1

3 2

2 3

x x x

x x x

x x

!

!

!

1 2

1 2

2 4

4 3 3

x x

x x

!

!

1 2

1 2 3

1 2 3

1

2 2 5

2 2 3

x x

x x x

x x x

!

!

!

1 2

3 4

Solve each of the following system of equality by Cramers Rule


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6.13 Solve Using Gauss &

Gauss-Jordan Elimination Method

For any matrix

Gauss Elimination Method

Reduce the augmented matrix [A|b] into row echelonform

Starting with the last nonzero row, use back-substitution to find X

Gauss-Jordan Elimination Method

Reduce the augmented matrix [A|b] into reduced rowechelon form [I|X]

? AWrite in Augmented matrix |AX b A b


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Examples 3 (Solve Using Gauss &

Gauss-Jordan Elimination Method)

1 2

1 2

3 2 6

5 4 8

x x

x x

!

!

1 2 3

1 2 3

1 2

2 2 1

3 2

2 3

x x x

x x x

x x

!

!

!

1 2

1 2

2 4

4 3 3

x x

x x

!

!

1 2

1 2 3

1 2 3

1

2 2 5

2 2 3

x x

x x x

x x x

!

!

!

1 2

3 4

Solve each of the following system of equality by Gauss &Gauss-Jordan Elimination Method


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Example 4 (Solve system of equation )

Use inverse matrix, Cramers Rule, and Gauss &Gauss-Jordan Elimination Method to solve thefollowing system of equation. Compare youanswer.

0

2 7

2

x y z

y z

x z

!

!

!


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6.2 System of NonLinear Equations

By the end of this topic, you should be

able to

Solve a System of NonLinear Equations usingsubstitution

Solve a System of NonLinear Equations using

elimination


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Solve a System of NonLinear

Equations

System of NonLinear Equations contains 1 ormore nonlinear equation.

The solution(s) represent the point(s) of intersection

(if any) of the graphs of the equations.

There is no general methodology Substitution, elimination or neither

If the system contains 2 variables & easy to graph(lines, quadratic (parabolas), hyperbolas, circles &ellipse), then graph them.


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Examples 5

(Solve system of NonLinear Equations)

2

3 2

2 0

x y

x y

!

!

2 2

2

3 2 0

1 0

x x y y

y yx

x

!

!

2 2

2

13

7

x y

x y

!

!

2 2

2

4x y

y x

!

!

1 2

3 4

Solve each of the following system of nonlinear equality


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6.3 System of Inequalities

By the end of this topic, you should be

able to

Graph an inequality

Graph a system of Inequalities


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Graph an Inequality

Replace the inequality symbol by an equal sign andgraph the resulting equation

If the inequality is strict, use dashes mark

If the inequality is non-strict, use a solid mark

In each of the regions, select a test point P

If the coordinate ofPsatisfy the inequality, then all the points in

that region satisfy the inequality. Indicate this by shading theregion

If the coordinate ofPdo not satisfy the inequality, then none ofthe points in that region do.


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Examples 6 (Graph an Inequality)

3 2x y

2 2x y "

4x y u

2 2x y e

1 2

3 4

Graph each of the following Inequality


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Graph a system of inequality

Graph each inequality in the system

Superimpose all the graphs

The overlapping regions are the

solutions of the system.

If there is no overlapping region, thesystem has no solution.


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Examples 7

(Graph a system of Inequality)

2

2 4

x y

x y

u

e

2 2

2 0

x y

x y

u

u

2

0

x y

x y

e

u

3

2 4

0

0

x y

x y

x

y

u

u

u

u

1 2

3 4

Graph each of the following system of Inequality

2 2

0

x y

x y

e

u

6 25

155

0

0

x y

xy

x

y

e

ue

u

u5


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