x+y+z=2, 2x-y+5z=-5,-x-2y+2z=1
x+y+z=2 <= eq1
2x-y+5z= -5 <= eq2
-x-2y+2z=1 <= eq3
eq1 + eq2
x + y + z = 2 add to 2x - y + 5z = - 5
3x + 6z = - 3
x + 2z = -1 <= eq4
eq1 + eq3
x + y + z = 2 add to - x - 2y + 2z = 1
y + 3z = 3 <= eq5
in eq4 : x = - 1 - 2z <= 6
eq6 subtitute to eq3
- x -2y+2z = 1 <= eq3
-(-1 - 2z) - 2y + 2z = 1
1 + 2z - 2y + 2z = 1
- 2y + 4z = 0
4z = 2y
2z = y
z = y/2 < eq7
eq7 in eq5
y + 3z = 3 <= eq5
y + 3(y/2) = 3
2y + 3y = 3(2)
5y = 6
y = 6/5
y in eq2
2x-y+5z= -5 <= eq2
2x - (6/5) + 5z = - 5
2x - (6/5) + 5z + 5 = 0
10x - 6 + 5(5)z + 5(5) = 0
10x + 25z +25 - 6 = 0
10x + 25z = -19 <eq8
x + 2z = -1 <= eq4
x = - 2z - 1 <= x into eq8
10(-2z - 1) + 25z = -19
- 20z + 10 + 25z = -19
+ 10 + 5z = -19
5z = - 19 - 10
5z = - 29
z = -29/5
What we have computed y = 6/5; z = -29/5 enter into eq1
x+y+z=2 <= eq1
x + (6/5) + (-29/5) = 2
x + (6/5) + (-29/5) - 2 = 0
[(5)x + 6 - 29 -(5)2] = 0
5x + 6 - 29 - 10 = 0
5x - 33 = 0
5x = 33
x = 33/5
Summary: x = 33/5 : y = 6/5: z = -29/5 ◄ Ans