x+y+z=0, x-y+2z=5, 3x+y+z=8 now solve for x,y,and z

Question

solving the system using x,y,and z .

Answer 1

Elimination method

Given equations are x + y + z = 0  ---> (1)

x - y + 2 z = 5  ----> (2)

3 x + y + z = 8---> (3)

To eliminate the y  value add  (1)&(2).

x + y + z = 0

x - y + 2 z = 5

____________________

2 x + 3 z = 5 ---> ( 4 )

To eliminate the y  value add (2)&(3).

x - y + 2 z = 5

3 x + y + z = 8

______________

4 x + 3 z = 13 ---> (5)

To eliminate the z value subtract the equations (4)&(5).

2 x + 3 z  = 5

4 x + 3 z = 13       ( change the symbols )

_____________

- 2 x = - 8

Cancell the same signs.

x = 4.

Substitute the x value in (4).

2 x + 3 Z = 5

2 ( 4 ) + 3 z = 5

8 + 3 z = 5

3 z = 5 - 8

3 z = - 3

z = - 1.

Substitute the x and z values in eq ( 1 ).

x + y + z = 0.

4 + y + ( -1 )=0

y + 3=0

y= - 3.

Therefore  x = 4 , y = - 3 , z = - 1.