Linear

SystemsSchool year 2014/2015

Systems

A system consists of a set of equations, for

which we ask what are the common solutions.

ax+by=c

a’x+b’y=c’

System Degree

It’s the product of the higher degrees of the

individual equations that make up the system.

EXAMPLE:

4x-y²=3

x5-y=10

2 * 5 = gr.10

So a linear system is of first degree.

How do you solve linear systems?

- Substitution method

- Comparison method

- Reduction method

- Cramer’s method

Substitution method

Derive a variable by an equation and replace it

in the other equation.

ESEMPIO:

2x-5y=7

x-3y=1

x=3y+1

2(3y+1)-5y=7

x=3y+1

6y+2-5y=7

x=16

y=5

Comparison method

Derive the same variable

from both equations

a’x + b’y = c’canonical

form

ax + by = c

x - y = 3

x + y = 9

x = y + 3

x = -y + 9

x = y + 3

y + 3 = -y + 9

x = y + 3

2y = 6

x = 6

y = 3

Example:

Reduction method

Reduction Method

It’s to add or subtract

the corresponding terms of the two equations

to obtain an equation with only one unknow.

EXAMPLE: PROCEDURE:1. Multiplie one or both of the equations for factors

non-zero, so that the coefficients of one of the

variables are equal to or opposite.

2. If the coefficients obtained in step 1 are equal,

subtract member to member the two equations; if

the coefficients are opposite, add member to

member; so we get an equation in one unknown.

3. Solve the equation in a single variable.

4. Replace the solution in one of the two original

equations.

Cramer’s method

The system must be written in canonical form:

ax+by=c

a'x+b' y=c'

Determinant calculation (D)

+

-

= +ab'-a'b

Multiply +a with b' and -a' with b

a b

D=

a' b'

Dx calculation

+

-

= +cb'-bc'

Multiply +c with b' and -c' with b

c b

Dx=

c' b'

-

Dy calculation

+

= +ac'-a'c

Multiply +a with c' and -a' with c

a c

Dy=

a' c'

If D ≠ 0 the

system is

determined

x=Dx/D

y=Dy/D

If D = 0

-the system is

indeterminate with

Dx and Dy=0

-the system is

impossible with Dx

and Dy≠0

Literal Systems

The literal systems are those where in

addition to the variables there are other

letters (parameters).

Example

Transform the system in canonical form.

2x= 2a-y 2x+y=2a

(a+1)x+ay=2a (a+1)x+ay=2a

For literal system the most used method is Cramer, calculating the determinant D, Dx, Dy.

D= = 2a-(a+1) = 2a-a-1= a-1 Dx= = 2a2 -2a=2a(a-1)

Dy= = 4a -2a(a+1)=4a-2a2 -2a=2a-2a2 =2a(1-a)

2 1

a+1 a

2a 1

2a a

2 2a

a+1 2a

...continuous example

The system is determined if D ≠ 0 ie if a-1≠0 a≠1.

● If a≠1 then

● If a=1 then D=0, Dx=0 and Dy=0 and the system is indeterminated.

x= Dx/D= 2a(a-1)/a-1= 2a

y=Dy/D= 2a(a-1)/a-1= -2a(a-1)/a-1= -2a

x= 2a

y= -2a

Linear Fractional Systems

When a system is fractional?

● Are those systems in which at least one of the equations that compose

it appears the unknown of first degree (x; y) in the denominator.

● Is solved with the methods we have already seen. (eg. the

replacement method; method of comparison; reduction method etc ...),

but it should be the

EXISTENCE CONDITION (E. C.)

steps shall be non-zero

all denominators that contain the unknown

Example

Found:

● l.c.m= 2xy

● E.C.: x≠0 U y ≠0

Transform the system in canonical form...

…and choose the most appropriate method to solve it.

Result:

Check if the solution of the system

satisfies the E.C.

Sistems with 3 equations and 3 variables

CANONICAL FORM

Solve operations in brackets

Order and simplify the terms like putting the system in CANONICAL FORM

CANONICAL FORM

Find the value of y will go out and replaced in the other two equations using the

method of substitution

So the coefficient of y of the first equation is equal to 1, derive the value of y

Solve in order to remove the brackets

Order and simplify the similar terms in the equations in which the value replaced

The first equation must be simplified for 5

Since the coefficient of z of the first equation is equal to 1, derive from it the

value of z

Found the value of z and replace in the other two equations using

the substitution method

Solve operations in brackets

Order and simplify opposite terms

Find the value of x in the first equation and substitute in last

The system solution is given by the triplet (4; 0 ; 5 ) that simultaneously

solves all of the system equations. The system is therefore DETERMINED .

WORK MADE BY THE CLASS 2nd A Afm

OF ITCG “CORINALDESI” - SENIGALLIA (AN) - ITALY

Team 1 - Breccia Martina, Franceschetti Sofia, Pinca Julia Andrea

Team 2 - Valentini Alessia, Esposto Giorgia, Biagetti Elena, Montironi Ilaria, Carletti

Lucia

Team 3 - Fabri Luca, Franceschini Simone, Bernardini Alessio, Urbinelli Riccardo,

Saramuzzi Mirko

Team 4 - Rossi Davide, Ventura Devid, Latini Angelo

Team 5 - Cervasi Michela, Casella Federica, Raccuja Ilaria

Team 6 - Zhang Qiuye , Zhang Ting, Xie Sandro

Team 7 - Trionfetti Sara, Carbonari Gloria, Borgacci Francesca, Avaltroni Alessia