Question

Given the equation x+y+z=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1 . Prove that at least one of x, y, z is equal to 1

Collected in the board: Algebraic proof questions

Steven Zheng posted 2 months ago


Answer

Remove denominator from the fraction

\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} =1

xy+yz+zx=xyz

Rearrange the terms

xyz-xy-yz-zx+1-1=0

Replace the first 1 with x+y+z based on given condition

xyz-xy-yz-zx+x+y+z-1=0

Factoring by grouping the terms

(x-1)(y-1)(z-1)=0

Therefore, in order for the equation to be valid, at least one of the variables x, y, z is equal to 1.

Steven Zheng posted 2 months ago

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