Designate all three (x, y, z)

DAXE

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Sep 9, 2019
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5
Please help in problem:
Determine all three (x, y, z) real numbers other than 0 for which the equation is met xy(x+y)=yz(y+z)=zx(z+x)
I have calculated that for sure x = y = z = 2 but I can't write it mathematically.
Thanks for all suggestions.
 
Determine all three (x, y, z) real numbers other than 0 for which the equation is met xy(x+y)=yz(y+z)=zx(z+x)
Suppose that \(\displaystyle \alpha\in\mathbb{R}\setminus\{0\}\), surely if \(\displaystyle \bf x=y=z=\alpha\) works?
 
Suppose that \(\displaystyle \alpha\in\mathbb{R}\setminus\{0\}\), surely if \(\displaystyle \bf x=y=z=\alpha\) works?
Thank you for your help but how to prove it? How do you demonstrate this by turning into equations.
 
Determine all three (x, y, z) real numbers other than 0 for which the equation is met xy(x+y)=yz(y+z)=zx(z+x)
I have calculated that for sure x = y = z = 2 but I can't write it mathematically.
Suppose that \(\displaystyle \alpha\in\mathbb{R}\setminus\{0\}\), surely if \(\displaystyle \bf x=y=z=\alpha\) works?
Thank you for your help but how to prove it? How do you demonstrate this by turning into equations.
If \(\displaystyle x=y=z=\alpha\ne 0\) then \(\displaystyle xy(x+y)=yz(y+z)=zx(z+x)=2\alpha^3\).
 
As I interpret the problem, we are to find all triples (x, y, z) that satisfy the equations. The hard part is to be sure that there are no solutions where x, y, and z are not equal. I haven't seen that yet.
 
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