Question

2x+3x+4z=26
X+y+2z=11
3x+2y+z=19

Answer 1

Resposta:

x = 3, y = 4 e z = 2

Explicação passo a passo:

2x+3y+4z=26

 x+  y+2z=11

3x+2y+  z=19

Para resolver o sistema vamos utilizar a Regra de Cramer:

$\displaystyle \left[\begin{array}{ccc}2&3&4\\1&1&2\\3&2&1\end{array}\right] .\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}26\\11\\19\end{array}\right]$

$\displaystyle \Delta = \displaystyle \left[\begin{array}{ccc}2&3&4\\1&1&2\\3&2&1\end{array}\right] =(2).(1).(1)+(3).(2).(3)+(1).(2).(4)-(3).(1).(4)-(1).(3).(1)-(2).(2).(2)=28+(-23)=5$

$\displaystyle \Delta x = \displaystyle \left[\begin{array}{ccc}26&3&4\\11&1&2\\19&2&1\end{array}\right] =(26).(1).(1)+(3).(2).(19)+(11).(2).(4)-(19).(1).(4)-(11).(3).(1)-(2).(2).(26)=228+(-213)=15$

$\displaystyle x = \frac{\Delta x}{\Delta} =\frac{15}{5} =3$

$\displaystyle \Delta y =\left[\begin{array}{ccc}2&26&4\\1&11&2\\3&19&1\end{array}\right] =(2).(11).(1)+(26).(2).(3)+(1).(19).(4)-(3).(11).(4)-(1).(26).(1)-(19).(2).(2)=254+(-234)=20$

$\displaystyle y = \frac{\Delta y}{\Delta} =\frac{20}{5} =4$

$\displaystyle \Delta z= \left[\begin{array}{ccc}2&3&26\\1&1&11\\3&2&19\end{array}\right] =(2).(1).(19)+(3).(11).(3)+(1).(2).(26)-(3).(1).(26)-(1).(3).(19)-(2).(11).(2)=189+(-179)=10$

$\displaystyle z = \frac{\Delta z}{\Delta} =\frac{10}{5} = 2$

Answer 2

Resposta:

Olá bom dia!

Seja A a matriz das incógnitas:

       2   3  4

A =   1    1   2

       3    2  1

Obtemos a determinante de A:

                2   3  4  | 2  3

Det A =     1    1   2 | 1   1  

                3    2  1  | 3  2

Det A = 2*1*1 + 3*2*3 + 4*1*2 - (3*1*1 + 2*2*2 + 4*1*3)

Det A = 2 + 18 + 8 - (3 + 8 + 12)

Det A = 28 - (23)

Det A = 5

Agora obteremos as determinantes das matrizes parciais, onde a coluna da variável a ser obtida é substituída pela coluna dos termos independentes.

26  3   4 | 26  3

11    1    2 | 11    1

19   2   1  | 19   2

Dx = 26*1*1 + 3*2*19 + 4*11*2 - (3*11*1 + 26*2*2 + 4*1*19)

Dx = 26 + 114 + 88 - (33 + 104 + 76)

Dx = 140 + 88 - (213)

Dx = 15

2  26   4 | 2   26  

1    11    2 | 1    11

3   19   1  | 3    19  

Dy = 22 + 156 + 76 - (26 + 76 + 132)

Dy = 20

2   3  26  | 2  3

1    1    11   | 1   1

3   2    19 | 3  2

Dz = 38 + 99 + 52 - (57 + 44 + 78)

Dz = 189 - 179

Dz = 10

Determinaremos agora os valores de x, y e z:

x = Dx / Det A

x = 15 / 5

x = 3

y = Dy  / Det A

y = 20 / 5

y = 4

z = Dz  / Det A

z = 10 / 5

z = 2

S = {(3 , 4 , 2)}