8x - y + z = -9, 2x + 2y - 3z = -22 and x - 3y + 2z = 15 solve for an ordered tripple

Question

8x - y + z = -9, 2x + 2y - 3z = -22  and x - 3y + 2z = 15.

Answer 1

Elimination method :

Given equations are 8 x - y + z = - 9  ---> (1)

2 x + 2 y - 3 z = - 22  ----> (2)

x - 3 y + 2 z = 15  -----> (3)

Multiply each side by - 4 in eq ( 2)

We get the equation -8 x -8 y + 12 z  =  88  ---------> ( 4 )

To eliminate the x  value add  (1)&(4).

8 x - y + z = - 9

- 8 x - 8 y + 12 z = 88

__________________

- 9 y + 13 z =  79 ---> ( 5 )

Multiply each side by - 2  in eq ( 3 )

We get the equation - 2 x +6 y - 4 z  = - 30 --------> ( 6 )

To eliminate the x  value add (2)&(6).

2 x + 2 y - 3 z = - 22

- 2 x +6 y - 4 z  = - 30

_________________

8 y - 7 z = - 52 ---> (7)

Multiply each side by  8  in eq (5 ) 

We get the equation - 72 y +104 z = 632 -------> ( 8 )

Multiply each side by  9 in eq (7 ) 

We get the equation 72 y - 63 z = - 468   -------> ( 9 )

To eliminate the x value add (8)&(9).

- 72 y +104 z = 632

72 y - 63 z = - 468 

______________

41 z =  164

z = 164 / 41 =4

z = 4

Substitute the z value in eq ( 5 ).

- 9 y + 13 z =  79

- 9 y +13 (4) = 79

-9 y +  52 = 79

- 9 y  = 79 - 52

- 9 y  =  27

y = - 3

Substitute the values x & y valuesin eq ( 1 )

8 x - y + z = - 9

8 x - ( - 3 ) + 4 = - 9

8 x + 3 + 4 = - 9

8 x  = - 9 - 7

8 x = - 16

x = - 2

Solution of the system of  the equations is x = - 2 , y  =  - 3 and  z = 4.

Comments

So ordered tripple for given system of equations is ( -2 ,-3 ,4)