Let x, y, z be non-zero real numbers, such that x + y + z = 0 and the numbers
x/y + y/z + z/x and x/z + z/y + y/x + 1
are equal. Determine their common value.
Given: x/y + y/z + z/x = x/z + z/y + y/x + 1
Given: x+y+z = 0, so say z=-(x+y)
Then: x/y+ y/(-x-y) + (-x-y)/x = x/(-x-y) + (-x-y)/y + y/x + 1
But since y is some multiple of x, say kx with k over the reals
x/(kx)+ (kx)/(-x-(kx)) + (-x-(kx))/x = x/(-x-(kx)) + (-x-(kx))/(kx) + (kx)/x + 1
Simplifying to k/(k + 1) + k = 1/k, giving some quite nasty values for k around -2.247, -0.55496, and 0.80194
However:
x/(-2.247x) + (-2.247x)/-(x+(-2.247x)) + -(x+(-2.247x))/x ≈ -1
x/(-0.55496x) + (-0.55496x)/-(x+(-0.55496x)) + -(x+(-0.55496x))/x ≈ -1
x/(0.80194x) + (0.80194x)/-(x+(0.80194x)) + -(x+0.80194x)/x ≈ -1
So the answer is that the two expressions are each worth -1
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Posted by broll
on 2021-08-06 00:19:13 |
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