Autor: Laurevânia | Publicado: 2012-10-24T12:23:00-07:00


SISTEMA
DE EQUAÇÕES DO 1º GRAU COM DUAS INCÓGNITAS







Da
localização de posições&nbsp;à resolução de problemas



Prof. Laurevânia





Quando os números se&nbsp;

vestem com letras




•Aritmética vem da
palavra grega arithmós, que significa número,
mas a maneira como escrevemos os números e com eles calculamos é herança dos
árabes, como, por exemplo, do matemático Al-Khowarizmi, que viveu por volta do ano 800 depois de Cristo.




•Álgebra vem de al-jabr, devido a Al-Khowarizmi .


“uma
certa regra para transformar uma igualdade numa outra igualdade tendo o mesmo
valor(isto é, equivalentes), regra simples e fácil de compreender”





Quando
queremos resolver problemas através da algébra



•Usamos
letras para representar números;

•Escrevemos
equações para traduzir as condições do problemas e trabalhamos com as letras
como se eles fossem números;

•Essa
é uma nova linguagem, uma “nova” matemática, a “Matemática das letras”





&nbsp;Acompanhe a
resolução algébrica abaixo e tire suas conclusões:



•A soma da minha idade com a de meu
irmão é 26. Dentro de dez anos, meu irmão terá o dobro da idade que tenho hoje.
Quais são as nossas idades? 



•Resolução: Vamos representar as
idades por duas letras.



&nbsp; Hoje

Minha idade _________________________ x

Idade de meu irmão ___________________ y

Idade de meu irmão daqui a dez anos
_____ y +
10



Agora, escrevemos duas equações
relacionando os dados do problema





&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;x
+&nbsp; y&nbsp;&nbsp;&nbsp;
=&nbsp; 26&nbsp; _____ &nbsp;A soma é&nbsp; 26

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;y&nbsp; + 10
=&nbsp; 2x&nbsp;&nbsp;
_____ &nbsp;Dentro de dez
anos, a

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;idade de meu irmão 

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;será o dobro da idade

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;que tenho hoje.&nbsp; 




&nbsp;&nbsp;&nbsp;&nbsp; Essas duas equações
formam um sistema de equações do 1º grau com duas incógnitas. Resolvendo
teremos x = 12 e&nbsp;&nbsp; y = 14, ou seja, eu
tenho 12 anos e meu irmão tem 14 anos.&nbsp;



SOLUÇÕES
DE UMA EQUAÇÃO DO 1º GRAU COM DUAS VARIÁVEIS





&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;•Toda
equação do 1º grau com duas incógnitas, x e y, por exemplo, tem infinitas
soluções, cada uma delas indicada por um par ordenado de números: o primeiro
representa sempre o valor da incógnita x, o segundo representa sempre o valor
da incógnita y. Essa ordem precisa ser respeita. Daí o nome par ordenado.
Indica-se (X, Y).


Vamos
verificar isso analisando os exemplo a seguir.





1 O par ordenado (2,5)
é a solução da equação&nbsp; 3x&nbsp; +&nbsp;
2y&nbsp; =&nbsp; 16?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3x&nbsp; +&nbsp;
2y&nbsp; =&nbsp; 16

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3. (2) + 2. (5)&nbsp; =&nbsp; 16

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6&nbsp; +&nbsp;
10&nbsp; =&nbsp; 16 (verdadeira)




2 O par ordenado (4,2)
é solução da equação 3x&nbsp; +&nbsp; 2y&nbsp;
=&nbsp; 16?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3x&nbsp; +&nbsp;
2y&nbsp; =&nbsp; 16

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3. (4)&nbsp; +&nbsp;
2.2&nbsp; =&nbsp; 16

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 12&nbsp; +&nbsp;
4&nbsp; =&nbsp; 16








MÉTODOS DE RESOLUÇÃO DOS SISTEMAS&nbsp;


&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;DO 1º GRAU COM DUAS INCÓGNITAS



Existem vários
métodos de resolução, entre os quais:

1)Método da substituição

&nbsp; &nbsp; &nbsp; Este método consiste em achar o valor
de uma das incógnitas em uma das equações e substituí-la na outra.


Exemplo:

Seja o sistema:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;x + y = 5&nbsp;

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;x - y = 1&nbsp;

Da primeira equação
podemos tirar que: x = 5 - y

Substituindo esse valor
na segunda equação:&nbsp;

x - y = 1

5 - y - y = 1

- 2y = 1 - 5

- 2y = - 4 &nbsp;.(-1)

2y = 4

y = 4/2

y = 2

Substituindo y por 2
em &nbsp;x + y = 5, temos:


x + 2 = 5

x = 5 - 2

x = 3


&nbsp; &nbsp; &nbsp; &nbsp; Logo: &nbsp;V = {(3, 2)}



&nbsp; &nbsp; &nbsp;2 ) Método de adição




&nbsp; &nbsp; &nbsp; &nbsp;Este método consiste na eliminação de
uma das incógnitas, adicionando-se membro a membro as duas equações. É
necessário que os coeficientes da incógnita que se deseja eliminar seja
simétricos.








&nbsp;Exemplo:

&nbsp;Seja o sistema:&nbsp; &nbsp; &nbsp; &nbsp;
&nbsp;&nbsp;x + y = 5&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;

&nbsp;&nbsp;x – y = 1










Somando -se membro a membro as duas equações:


&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; x + y = 5&nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; x – y = 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2x &nbsp; &nbsp;= 6

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x = 6/2

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;x = 3

Substituindo-se esse valor de x em uma das equações&nbsp;dadas:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;


&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x + y = 5

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3 &nbsp;+ y = 5

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;y = 5 - 3

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;y = 2 &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Logo: V = {(3,2)}









&nbsp;Exemplo 2

Seja o sistema:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 4x – y&nbsp; = 2

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3x + 2y
= 7




•Nesse caso, não temos coeficientes simétricos. Vamos,
então, multiplicar todos os termos da primeira&nbsp;
equação por 2.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp; 4x – y =
2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;

&nbsp;&nbsp;8x – 2y = 4 &nbsp;



&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;8x – 2y = 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3x + 2y = 7

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;11x&nbsp; &nbsp; &nbsp;&nbsp;= 11

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;x = 11/11

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;x&nbsp; = 1&nbsp;




-Vamos substituir este
valor de x em uma das equações dadas (por&nbsp;exemplo a segunda):&nbsp;




3x + 2y = 7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2y = 4

3.(1) + 2y = 7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;y = 4/2

3&nbsp; + 2y = 7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;y = 2

2y = 7 – 3 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp;Logo: V= {(1 , 2)}

&nbsp; &nbsp;&nbsp;



Exemplo 3:


Seja o sistema:&nbsp;

4x + 2y =
16

5x
– 3y = 9





•4x+2y =16[vamos multiplicar essa equação por 3]=12x = 6y= 48

•5x–3y = 9[vamos multiplicar essa equação por&nbsp;2]=10x – 6y= 18





Observe: 

•Somando membro a membro as equações:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 12x + 6y = 48

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 10x – 6y = 18

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 22x&nbsp;&nbsp;= 66

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;x = 66/22&nbsp;&nbsp;&nbsp;x = 3





- Substituindo-se
esse valor por x em uma das equações dadas (por exemplo, na primeira)

4x + 2y = 16&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2y = 16 - 12&nbsp; &nbsp; &nbsp;&nbsp;

4 .3 + 2y = 16&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2y = 4

12 + 2y = 16&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;y = 4/2

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;y = 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;


Logo: V = {(3, 2)}