domingo, 19 de abril de 2020

SISTEMAS DE EQUAÇÕES DO 1° GRAU




Colégio Estadual Dinah Gonçalves
email accbarroso@hotmail.com
www.youtube.com/accbarroso1  

Existem vários métodos de resolução entre os quais:


1) MÉTODO DA SUBSTITUIÇÃO

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



EXEMPLO 1

Seja o sistema

X + Y = 5

X - Y = 1



Da primeira equação podemos tirar que:

x + y = 5 sendo assim passando o y para o outro lada do igual e invertendo os sinais fica: x= 5-y

já que x vale ou é igual (5 -y) substituindo o valor de x na outra equação do sitema temos :

X – y = 1
(5 –y) – y = 1
-y –y = 1 -5
-2y= -4
y = -4 / -2
y= 2

Substituindo y por 2 em x = 5 – y
____________________x = 5 -2
____________________x = 3

portando o resultando do sistema é ( 3,2)


EXEMPLO 2

Seja o sistema

X – 2y = 3

2x – 3y = 5

Sendo assim da primeira equação tiramos

X – 2y = 3 __________ x = 3 + 2y

Substituindo o valor de x na segunda equação :

2x – 3y = 5
2(3 + 2Y) – 3y = 5
6 + 4y – 3y = 5
4y – 3y = 5 – 6
y = -1

Substituindo y por -1 em :
x = 3 + 2y
x = 3 + 2 (-1)
x = 3 – 2
x = 1

logo a solução é ( 1 , -1)


2) MÉTODO DA ADIÇÃO

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 sejam simétricos .



EXEMPLO 1

Seja o sistema

X + y = 5

x – y = 1

Somando-se membro a membro as duas equações:

x + y = 5

x – y = 1
-----------
2x = 6

x= 6/2

x= 3

Substituindo esse valor de x em uma das equações dadas ( por exemplo na primeira)

x + y = 5
3 + y = 5
y= 5 – 3

y = 2

logo a solução é : (3,2)


EXEMPLO 2


Seja o sistema

4x - y = 2

3x + 2y = 7

Neste caso, não temos coeficientes simétricos. Vamos então multiplicar todos os termos da primeira equação por 2:

8x - 2y = 4

3x + 2y = 7
-----------
11x = 11

x = 11/11

x = 1

Vamos substituir este valor de x em uma das equações dadas (por exemplo, na segunda):

3x + 2y =7
3.1 + 2y + 7
3 + 2y = 7
2y = 7-3
2y = 4
y = 4/2

y = 2

Solução (1,2)

EXEMPLO 3


Seja o sistema

4x + 2y = 16

5x - 3y = 9


4X + 2Y = 16 ( vamos multiplicar essa equação por 3)
5x – 3y = 9 ( vamos multiplicar essa equação por 2)

Observe
Somando membro a membro as equaçãos

12x + 6y = 48

10x - 6y = 18
----------------
22x = 66

x= 66/22
x = 3

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

4x + 2y = 16
4.3 + 2.y= 16
12 + 2y = 16
2y = 16 – 12
2y = 4
y = 4/2
y = 2

solução (3,2)



EXERCICIOS


A) Calcule os sistemas


1) x - 3y = 1
_2x +5y = 13________ (R:4,1)


2) 2x + y = 10
__x + 3y = 15________ (R:3,4)

3) 3x + y = 13
__2x - y = 12________ (R:5,-2)

4) 2x + 7y = 17
__5x - y = -13________ (R:-2,3)

5) 2x + y = 4
__4x - 3y = 3________ (R:3/2,5)

6) x + y = 2
_3x + 2y = 6________ (R:2,0)

7) x/2 + y/3 = 3
____x - y = 1________ (R:4,3)

8) x - y =5
__x +y = 7________ (R:6,1)

9) x - y =2
_2x +y = 4________ (R:2,0)

10) x + y =3
__2x +3y = 8________ (R:1,2)

11) x - 3 = 0
__2x - y = 1________ (R:3,5)

12) 3x + y =5
___2x +y = 4________ (R:1,2)

13) x = y - 2
__2x +y = -1________ (R:-1,1)

14) x - y -2 = 0
__2x +y – 7= 0________ (R:3,1)

15) x + y = 7
___x -y = 1________ (R:4,3)

16) x + y = 6
__2x +y = 4________ (R:-2,8)

17) 2x + y = 5
___x + 2y = 4________ (R:2,1)

18 ) x + y = 11
___x - y = 3________ (R:7,4)

19) x - y = 16
___x +y = 74________ (R:45,29)

20) x - y = 1
___x +y = 9________ (R:5,4)

21) 2x - y = 20
___2x +y = 48________ (R:17,14)

22) x + y = 1
___x - y = 7__________ (R:4, -3)

23) x + y = 3
___x - y = -5_________ (R:-1,4)

24) x + y = 5
___x- y = -5_________ (R: 0,5)

25) Se x e y é a solução do sistema

x + y = 4
x+ 2y = 6

então x - y é:
a) 8
b) 6
c) 4
d) 2
e) 0 (X)


26) Se x e y é a solução do sistema
a + b = 3
2a+ b = 5

então a - b é:
a) 0
b) 2
c) 4
d) 3
e) 1 (X)
27) Qual a solução do sistema de equações abaixo?

x – y = 3
2x + y = 9

a)(1,0)
b)(2,3)
c)(3,2)
d)(4,1) (X)e)(5,3)

28) A solução do sistema

2x + y = 10
x + 3y = 15 é

a) x=3 e y=4 (X)
b) x=3 e y=5
c) x=2 e y=4
d) x=1 e y=5
e) x=5 e y=3

29) Se x e y é a solução do sistema

x + 3y = 9
3x+ 2y = 6

então x - y é:

a) 0 (X)
b) 3
c) 6
d) 9











B) RESOLUÇÃO DE PROBLEMAS COM SISTEMAS

1) Determine dois números, sabendo que sua soma é 43 e que sua diferença é 7 (R:25,18)

2) Um marceneiro recebeu 74 tabuas de compensado. Algumas com 6 mm de espessura e outras com 8 mm de espessura. Quando foram empilhadas atingiram uma altura de 50 cm. Quanta tabua de 8mm ele recebeu? (R: 28)

3) Em um estacionamento havia carros e motocicletas num total de 43 veículos e 150 rodas. Calcule o numero de carros e de motocicletas estacionadas. (R:32,11)

4) Uma empresa deseja contratar técnicos e para isso aplicou um prova com 50 perguntas a todos os candidatos. Cada candidato ganhou 4 pontos para cada resposta certa e perdeu um ponto para cada resposta errada. Se Marcelo fez 130 pontos quantas perguntas ele acertou? (R: 36)
5) Pedro e Paulo tem juntos R$ 81,00. Se Pedro der 10% do seu dinheiro a Paulo eles ficarão com quantias iguais. Quanto cada um deles tem? (R: 45,36)

6) Descubra dois números inteiros que somados dão 88, sabendo que um é igual ao triplo do outro (R:66,22)

7) Num quintal há 100 animais entre galinhas e coelhos. Sabedo que o total de pés é 320, quantas galinhas e quantos coelhos há nesse quintal? (R 40,60)

8) Num estacionamento há 80 veiculos, entre motos e carros. Se o total de rodas é 190, quantos carros e quantas motos há nesse estacionamento? ( R:65,15)

9) Um teste é composto de 40 questões. Para cada questão respondida certa são atribuidos três pontos (+3) Para cada questão respondida errada são descontados dois pontos (-2) Ilda respondeu a todas as questões desse teste e fez um total de 75 pontos . quantas questões foram respondidas certas? ( R: 31)

10) Um caminhão carrega 5000 pacotes de açucar de 2 kg e de 5 kg num total de 15 400 kg. Quantos pacotes de 2 kg e 5 kg esse caminhão está transportando ? (R: 3200,1800)

11) Ache dois números que a soma deles é 354 e a diferença entre eles é 128. ( R: 241,113)


SISTEMAS SIMPLES DO 2° GRAU

Vamos resolver sistemas que possuem uma equação do 1° grau e outra do 2° grau polo método da substituição .

Exemplo

a)x - y = 1
__x . y = 6

Isolando x na equação x - y = 1, temos x = 1 + y
substituindo esse valor de x em x . y = 6 , obtemos,
(1+ y) . y = 6
y + y² = 6
y² + y - 6 = 0

resolvendo essa equação do 2° grau temos:

solução ( 2 e -3)

b) x + y = 7
___x². y² = 25

Isolando x na equação x + y = 7, temos x = 7 - y
substituindo esse valor de x em x². y² = 25, obtemos

(7 - y)² + y² = 25
49 - 14y + y² + y² = 25
2y² - 14y + 49 - 25 = 0
2y² - 14y + 24 = 0

resolvendo essa equação do 2° grau temos:

solução [(3 e 4) e (4 e 3)]

Exercícios


1)x + y = 7______ (R:2,5 ou 5,2)
__x . y = 10

2)x + y = 5______ (R:2,3 ou 3,2)
__x . y = 6

3)x - y = 9______ (R:2,-7 ou -7,2)
__x . y = -14

4) x - y = 3______ (R:6,3 ou -3,-6)___x²+ y² = 45

5) x - y = 1______ (R: 4,3 ou -3,-4)
___x²+ y² = 25

b) x - y = 0______ (R: 2,2 ou -2,-2)
___5x²- y² = 16
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110 comentários:

  1. como resolver a 10 questão do sistema?

    ResponderExcluir
    Respostas
    1. Na linguagem matematica encontramos o seguinte sistema 2x+5y=15400
      x+y=5000

      multiplicando a de baixo por (-2) vamos obter o equivalente

      2x+5y=15400
      -2x-2y=-10000 resolvemos pelo metodo da soma y=1800
      x=3200

      Excluir
    2. Na linguagem matematica encontramos o seguinte sistema 2x+5y=15400
      x+y=5000

      multiplicando a de baixo por (-2) vamos obter o equivalente

      2x+5y=15400
      -2x-2y=-10000 resolvemos pelo metodo da soma y=1800
      x=3200

      Excluir
    3. 10)
      x + y =3 SUBSTITUINDO ~> x = 3 - y
      2x + 3y = 8
      2(3 - y) + 3y = 8
      6 - 2y + 3y = 8
      y = 8 - 6
      y = 2

      x = 3 - y
      x = 3 - 2
      x = 1 S = {1 , 2}

      Excluir
  2. cara onde estao as respostas sao os numeros de vermelho e

    ResponderExcluir
  3. Caro amigo....

    Nos cálculos de sistema você errou a questão 5.
    O Correto é: (4,5 e -5) e Assim você confunde o aluno.
    Corrija por favor!

    ResponderExcluir
    Respostas
    1. Na primeira questão, letra A, ele também errou deu {-44;-15}

      Excluir
    2. Na primeira questão, letra A, ele também errou deu {-44;-15}

      Excluir
    3. A letra A dá certo sim,
      {x-3y=1
      {2x+5y=13

      x-3y=1
      x=1+3y
      2(1+3y)+5y=13
      2+6y+5y=13
      6y+5y=13-2
      11y=11
      y= 11
      ----
      11
      y= 1


      2x+5y=13
      2x=5(1)=13
      2x+5=13
      2x=13-5
      2x=8
      {(x=8/20)}=4

      S{(4;1)}

      Excluir
    4. Enunciado está como-2×+5y=13

      Excluir
    5. Este comentário foi removido pelo autor.

      Excluir
  4. Este comentário foi removido pelo autor.

    ResponderExcluir
  5. Não coloca as respostas na frente da questão não, amigo!! Atrapalha muito quem quer fazer, na próxima coloque o gabarito no final, ficaria melhor.

    ResponderExcluir
  6. Este comentário foi removido pelo autor.

    ResponderExcluir
  7. Este comentário foi removido pelo autor.

    ResponderExcluir
  8. a resposta da questão 5 da A ta errada

    2x+y=4
    4x-3y=3

    2x=4-y

    4x-3y=3
    2(4-y)-3y=3
    8-2y-3y=3
    -5y=-5 (-1)
    5y=5
    y=5/5
    y=1

    2x=4-y
    2x=4-1
    2x=3
    x=3/2
    x=1,5

    ResponderExcluir
    Respostas
    1. fiz pelo método da adição e deu a mesma coisa, e ai eu vi que tem alguma coisa errada fiz e refiz e não da essa reposta do enunciado da questão!

      Excluir
  9. Boa tarde! Eu resolvi a questão 5, e a resolução foi: ( 3\2,1) e não é 3\2,5). Tentei de várias formas achar o valor de Y para q seja 5, porém não deu. Ai substitui o valor de Y q é 1 nas duas equações e bateu com resultado.

    ResponderExcluir
  10. Qual a resolução (passo a passo) da questão 29, por favor???!!!

    ResponderExcluir
    Respostas
    1. vamos fazer pelo metodo da adiçao x-3y=9 multiplica por -3
      entao vai ficar -3x-9y=-27
      a outra equaçao e 3x+2y=6 elimina o x que no caso e 3 e soma as outras partes do sistema entao vai ficar -7y=-21 e multiplica por menos 1 que o y vai ficar 3
      depois substitui y por 3 na 2 equaçao 3x+3.2=6 3x+6=6 3x=0 x=3 x-y=3-3=0 fuiiiiii

      Excluir
    2. Esta errado! Na equação 2 ao substituir vc encontra: 3x+2.3=6. Realizando os cálculos fica 3x + 6 = 6. Após isso ficará 3x=6-6 e 3x=0. Até ai vc acertou. So que ao isolar x voce encontra 0 que divide 3. Resultado é x=O.
      E ao realizar o que foi pedido de x - y voce fica com 0 - 3= -3. Não tem essa opção.

      Excluir
  11. na hora de substituir o y e o resultado da, -y=2-4x, você tem de multiplicar por -1, porque não tem como ser -y=2-4x.

    ResponderExcluir
  12. galera o numero 5 da questão A o y só ta dando 1 não to conseguindo achar esse 2,5 pro y fiz ele no metodo de adição pois os outros não estudei ainda :/ alguém ai deu conta?

    ResponderExcluir
  13. A resposta do exercicio 7 da Questão A está errado ao meu ver. A minha solução está dando 5 e 4, e bateu com as duas equações (obs: fiz utilizando tanto o método de adição e susbtituição)

    ResponderExcluir
  14. Outra dúvida galera. Dependendo do método usado, pode dar resultados diferentes?

    ResponderExcluir
  15. eu sem querer usei ao invés de 15400, usei apenas 400 e a conta deu certo kkkk

    ResponderExcluir
  16. ALGUEM ME AJUDA POR FAVORRRR !!!!!!!!!!!!!!

    2 ( X+1 ) = 3 (Y+2) QUESTAO 1
    2X - 2Y = -8



    ME AJUDEM ENVIA PARA O NUMERO DO WHATS OU RESPONDE=ME AQUI MESMO NUMERO -> 69 9 9301 8288

    ME AJUDEM POR FAVORRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR

    ResponderExcluir
    Respostas
    1. Spakay Br
      x+y=43
      x-y=7

      X=43-y
      (43-y)-y=7
      43y+y=7
      Y+y=7-43
      2y=36
      Y=36/2
      Y=18

      X-y=7
      X-18=7
      X=7+18
      X=25

      Excluir
  17. Spakay Br
    x+y=43
    x-y=7

    X=43-y
    (43-y)-y=7
    43y+y=7
    Y+y=7-43
    2y=36
    Y=36/2
    Y=18

    X-y=7
    X-18=7
    X=7+18
    X=25

    ResponderExcluir
  18. Qual ea resposta da 1 2 3 4 5 6 7 8 9 10 11 12 13 14 e da 20 não to dano conta

    ResponderExcluir
  19. se 2 vezes d 2 e 5 entao x equivale a potencia do infinito pois os vingadores mostra que o infinito e a disma certo?entao essas equaçoes tudo equilvale a um numero com virgula!!!

    obg sou professor de bob meu cachorro e estudei na escola chamada politica

    ResponderExcluir
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