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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