Question

REsolva os sistemas lineares:
a) x+y+3z=4
x+2y+z=5
2x+y+z=6



b) x-y+z=4
2x+3y-5z=-11
3x-2y+z=7

Answer 1

a)
x + y + 3z = 4
x + 2y + z = 5
2x + y + z = 6

D = 

[1   1   3     1    1 ]
[1   2   1     1    2]
[2   1   1     2    1]

D = 2 + 2 + 3 - (12 + 1 + 1)
D = 7 - (14)
D = - 7

Dx = 

[4  1  3  4   1   ]
[5  2  1  5   2  ]
[6  1  1  6   1  ]

= 8 + 6 + 15 - (36 + 4 + 5)
= 29 - (45)
= 29 - 45
Dx = - 16

x = dx  =  - 16 =  16
      d        - 7        7

x = 16/7
------------------------------------------------------------
Dy = 

[1   4  3   1   4]
[1   5  1   1   5]
[2   6  1   2   6]

= 5 + 8 + 18 - (30 + 6 + 4)
= 31 - (40)
= 31 - 40
= - 9

y = dy  =   - 9   =   9
       d       - 7         7

y = 9/7
---------------------------------------
Dz = 

[1  1   4    1    1]
[1  2   5    1    2]
[2  1   6    2   1]

= 12 + 10 + 4 - (16 + 5 + 6)
= 26 - (27)
= 26 - 27
= - 1

z = Dz  =    - 1  =  1
       D        -  7      7

z = 1/7

R.: x =  16/7, y = 9/7; z = 1/7
----------------------------------------------------------
b)
x - y + z = 4
2x + 3y - 5z = -11
3x - 2y + z = 7

D = 

[1    - 1    1       1     - 1]
[2     3    - 5      2       3]
[3   - 2     1       3     - 2]


D = 3 + 15 - 4 - (9 + 10 - 2)
D = 14 - ( 17)
D = 14 - 17
D = - 3
------------------------------------------------------------
Dx = 

[   4     - 1     1      4       - 1]
[- 11      3    - 5    - 11       3]
[  7       - 2      1       7     - 2]

Dx = 12 + 35 + 22 - (21 + 40 + 11)
Dx = 47 + 22 - (61 +11)
Dx = 69 - ( 72)
Dx = 69 - 72
Dx = - 3

x =  Dx  = - 3  =   1
        D      - 3

x = 1
---------------------------------------------------------------------
[1      4     1      1     4]
[2   - 11   - 5     2   - 11]
[3      7      1     3     7]

Dy = - 11 - 60 + 14 - ( - 33 - 35 + 8)
Dy = - 71 + 14 - (- 68 + 8)
Dy = - 57 - ( - 60)
Dy = - 57 + 60
Dy = 3

y = Dy =  3  =  - 1
      D     - 3

y = - 1
==========================
Dz = 

[1   - 1      4     1    - 1]
[2     3   - 11    2     3  ]
[3   - 2     7      3     - 2]

Dz = 21 + 33 - 16 - (36 + 22 - 14)
Dz = 54 - 16 - (36 + 8)
Dz = 38 - 44
Dz = - 6

z = Dz = - 6  =  2
       D     - 3

z = 2

R.: x = 1; y = - 1; z = 2

Answer 2

Letra A segue na foto em anexo.


Letra B


$\begin{cases}x-y+z=4\\2x+3y-5z=-11\\3x-2y+z=7\end{cases}\\\\\\\\ A= \left[\begin{array}{ccc}1&-1&1\\2&3&-5\\3&-2&1\end{array}\right]$




$D= \left[\begin{array}{ccccc}1&-1&1&1&-1\\2&3&-5&2&3\\3&-2&1&3&-2\end{array}\right] \\\\\\ (1.3.1)+(-1.(-5).3)+(1.2.(-2)-(3.3.1)-(-2.(-5).1)-(1.2.(-1))\\\\D= 3+15-4-9-10+2\to\\\\ D=20-23\to\\\\ \boxed{D=-3}$




$D_x= \left[\begin{array}{ccccc}4&-1&1&4&-1\\-11&3&-5&-11&3\\7&-2&1&7&-2\end{array}\right]$

Dx = (4.3.1)+(-1.(-5).7)+(1.(-11).(-2))-(7.3.1)-(-2.(-5).4)-(1.(-11).(-1)) --->

$D_x= 12+35+22-21-40-11\to\\\\ D_x=69-72\to\\\\ \boxed{D_x=-3}$




$D_y= \left[\begin{array}{ccccc}1&4&1&1&4\\2&-11&-5&2&-11\\3&7&1&3&7\end{array}\right] \\\\\\ (1.(-11).1)+(4.(-5).3)+(1.2.7)-(3.(-11).1)-(7.(-5).1)-(1.2.4)\\\\D_y = -11-60+14+33+35-8\to\\\\ D_y=-79+82\to\\\\ \boxed{D_y=3}$




$D_z= \left[\begin{array}{ccccc}1&-1&4&1&-1\\2&3&-11&2&3\\3&-2&7&3&-2\end{array}\right]$

(1.3.7)+(-1.(-11).3)+(4.2.(-2))-(3.3.4)-(-2.(-11).1)-(7.2.(-1)) --->

$D_z= 21+33-16-36-22+14\to\\\\ D_z=68-74\to\\\\ \boxed{D=-6}$





$x= \dfrac{D_x}{D} \to~~~~ x= \dfrac{-3}{-3} \to~~~~ \large\boxed{x=1} \\\\\\ y= \dfrac{D_y}{D} \to~~~~ y= \dfrac{3}{-3} \to~~~~ \large\boxed{y=-1} \\\\\\ z= \dfrac{D_z}{D} \to~~~~ z= \dfrac{-6}{-3} \to~~~~ \large\boxed{z=2} \\\\\\ \Large\boxed{\boxed{S=\{1~;~-1~;~2\} }}$