solve for , 2x-y+z=4, 4x+y-z=14 and z+3y-z=11

Question

2x-y+z=4

4x+y-z=14

z+3y-z=11

Answer 1

Elimination method :

Given equations are  2 x -y + z = 4 ---> (1)

4 x + y - z =14 ------> (2)

z + 3 y -  z = 11-----> (3)

Multiply each side by - 2  in eq ( 1)

We get the equation - 4 x + 2 y -2 z  = - 8 ---------> ( 4 )

To eliminate the x  value add  (2 )&(4).

4 x + y - z =14

- 4 x + 2 y -2 z  = - 8

________________

3 y - 3 z  = 6

y  -  z  =  2 -------> (5)

z + 3 y -  z = 11

3 y = 11

y = 11 / 3

Substitute the y  value in eq (5 ).

y  -  z  =  2

11 /3 -  z = 2

(11 / 3) - 2 = z 

5 /3 = z

Therefore z = 5 / 3

Substitute the values z  & y values in eq ( 1 )

2 x - y + z = 4

2 x - 11/3 + 5/3 = 4

2 x - 6/3 = 4

2 x -  2 = 4

2 x = 4 + 2

2 x = 6

x = 3. 

Solution of the system of  the equations is x = 3 , y =  11 / 3  and  z = 5 / 3.