1) 5x + 2y - 3z = -12
2) x - 4y + 2z = 23
3) 3x + y - 7z = -22
multiply eq(1) by 2, add to eq(2):
10x + 4y - 6z = -24
x - 4y + 2z = 23
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11x - 4z = -1 (a)
multiply eq(3) by 4, add to eq(2):
12x + 4y - 28z = -88
x - 4y + 2z = 23
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13x - 26z = -65 (b)
solve for x in eq(a):
11x = 4z - 1
x = (4z - 1)/11
substitute into eq(b):
13((4z - 1)/11) - 26z = -65
(52z - 13)/11 - 286z/11 = -715/11
52z - 13 - 286z = -715
-234z = -702
z = -702/-234 = 3
using eq(a):
11x - 4(3) = -1
11x - 12 = -1
11x = 11
x = 1
using eq(2):
1 - 4y + 2(3) = 23
-4y = 16
y = -4
check using eq(1):
5(1) + 2(-4) - 3(3) = -12
5 - 8 - 9 = -12
-12 = -12
x = 1, y = -4, z = 3 or (1,-4,3)