If $a^{x - 1} = b c, b^{y - 1} = a c, c^{z - 1} = a b$ , then find the value of xy + yz + zx - xyz.

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Solution

a^x / a = bc , a^x = abc
b^y/ b = ca , b^y = abc
c^z / c = ab . c^z = abc
a = (abc)^1/x
b = (abc)1/y
c = (abc)1/z
abc = (abc)^{(1/x+1/y+1/z)}
$(a b c) = (a b c)^{(} (x y + y z + x z) / x y z)$
(xy + yz + zx) / xyz = 1
xy + yz + zx - xyz=0

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