solve for x ,y, z, 3x-2y+2z=1, 2x+5y-5z=7 and 4x-3y+z=-3

Question

systems of equations containing three variables

Answer 1

Elimination method

Given equations are 3x-2y+2z = 1

Multiple to each side by 2.

6x-4y+4z = 2 -----> (1)

2x+5y-5z = 7

Multiple to each side by 3.

6x+15y-15z = 21 ----> (2)

4x-3y+z = -3 ----> (3)

To eliminate x variable, subtract equation (2) from (1).

6x+15y-15z = 21

6x-4y+4z    = 2

(-)  (+)  (-)  (-)

______________

19y-19z = 19

Divide to each side by 19.

y-z = 1 ----> (4)

2x+5y-5z = 7

Multple to each side by 2.

4x+10y-10z = 14 -----> (5)

To elminate the x variable, Subtract equation (5) from (3).

4x-3y+z = -3

4x+10y-10z = 14

(-)  (-)   (+)      (-)

________________

-13y+11z = -17 -----> (6)

Multiple to each side of equation (4) by 13.

13y-13z = 13 ----> (7)

To eliminate the y variable,add the equations (6)&(7).

-13y+11z = -17

13y-13z   = 13

___________

-2z = -4

z = 2

Substitute the z value in (4)

y-2 = 1

Add 2 to each side.

y = 3

Substitute z,y in (3).

4x-9+2 = -3

4x-7 = -3

Add 7 to each side.

4x = 4

Divide to each side by 4.

x = 1

Solution x = 1, y = 3, z = 2.