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Algebra Level 2

Is it true, that if x y + z = x z + y = y z + x xy+z=xz+y=yz+x then ( x − y ) ( y − z ) ( z − x ) (x-y)(y-z)(z-x) is equals to 0 for sure?

True False

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Chew-Seong Cheong
Jul 29, 2017

Let x y + z = z x + y = y z + x = k xy+z=zx+y=yz+x = k . Then, we have:

{ x = k − y z y = k − z x z = k − x y \begin{cases} x = k-yz \\ y = k - zx \\ z = k - xy \end{cases}

⟹ x − y = k − y z − k + z x = z ( x − y ) \implies x-y = k-yz - k +zx = z(x-y) . For the equation to be true for all value of z z , x − y = 0 x-y=0 and the product ( x − y ) ( y − z ) ( z − x ) = 0 (x-y)(y-z)(z-x) = 0 .

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Thanks for sharing your solution!

Áron Bán-Szabó - 3 years, 10 months ago

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