1) x + 2y + z = 10
2) 2x - y + 3z = -5
3) 2x - 3y - 5z = 27
multiply eq(1) by -2, add to eq(2):
-2x - 4y - 2z = -20
2x - y + 3z = - 5
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-5y + z = - 25 (a)
multiply eq(2) by -1, add to eq(3):
-2x + y - 3z = 5
2x - 3y - 5z = 27
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-2y - 8z = 32 (b)
multiply eq(a) by 8, add to eq(b):
-40y + 8z = -200
-2y - 8z = 32
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-42y = -168
y = -168/-42 = 4
using eq(a):
-5(4) + z = -25
z = -5
using eq(1):
x + 2(4) + (-5) = 10
x + 3 = 10
x = 7
check using eq(3):
2(7) - 3(4) - 5(-5) = 27
14 - 12 + 25 = 27
27 = 27
x = 7, y = 4, z = -5 or (7, 4, -5)