How Do You Do Systems With Three Variables?

X-2y+3z=-3 -------- eq- 1
2x-3y-z=7----------eq-2
3x+y-2z=6 --------eq-3

Multiply eq—3 by 3
9x + 3y – 6z = 18 ----------eq-4
Add eq-2 and eq-4
9x + 3y – 6z = 18
2x  -  3y – z  =  7
11x -7z = 25 -------eq--5
Multiply eq—3 by 2
6x + 2y – 4z = 12 -------eq--6
Add eq-1 from eq-6
6x + 2y – 4z = 12
x - 2y+3z= -3
7x – z =9 ------eq-7
Multiply eq—7 by 7
49x – 7z = 63------eq-8
Subtract eq-5 from eq-8
49x – 7z = 63
-11x +7z =- 25
38 x=38
X=1
Putting value of x in eq- 5
11x -7z = 25
11 – 7z = 25
-7z = 25 -11
-7 z = 14
Z= -2
Putting value of x= 1 and z= -2 in eq—1  we get
X-2y+3z=-3
1 – 2y + 3 (-2) = -3
1 – 2y -6 =-3
-2y – 5 = -3
-2y =-3 +5
-2y = 2
Y = -1

So Value of x = 1, y= -1 and z= -2