8x-y+z=-46, 3x+2y-3z=-18,x-3y+2z = 4

Question

solving system of equations, need to find all 3 variables

Answer 1

 8x-y+z = -46 -----> (1)

3x+2y-3z = -18 ----> (2)

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

Multiple to each side of  (3) by negitive 8⇒ -8x+24y-16z = -32 ----> (4)

Multiple to each side of  (3) by negitive3⇒ -3x+9y-6z = -12 -----> (5)

To elminate x value, add  equations (1) and (4).

  8x-y+z = -46

-8x+24y-16z = -32

________________

23y-15z = -78 ----> (6)

To elminate x value, add   (2) and (5).

3x+2y-3z = -18

-3x+9y-6z = -12

_____________

11y-9z = -30 -----> (7)

165y-135z = -450 ------> (8)

Multiple to each side of  (6) by negitive 9.

-207y+135z = 702 -----> (9)

To elminate z value, add  equations (8) and (9).

165y-135z = -450

-207y+135z = 702

_______________

-42y = 252        

-42y/-42 = 252/-42

⇒ y = -6

Substitute y value, in  (2).

3x+2*-6-3z = -18

3x-12-3z = -18  

3x-12+12-3z = -18+12

3x-3z = -6 ------> (10)

.Substitute y value, in  (3).

x-3*-6+2z = 4

x+18+2z = 4     

x+18-18+2z = 4-18

x+2z = -14 ------> (11)

Multiple to each side of  (11) by negitive 3.

-3x-6z = 42 -------> (12)

To elimnate the x value add   (10) and (12).

3x-3z = -6

-3x-6z = 42

__________

-9z = 36          

-9z/-9 = 36/-9

⇒ z = -4

Substitute y,z values, in  (2).

3x+2*-6-3*-4 =-18

3x = -18

3x/3 = -18/3

⇒ x = -6