crossoverman9b

2022-06-22

How many solutions are there to this equation?
$\begin{array}{rl}{x}^{2}-{y}^{2}& =z\\ {y}^{2}-{z}^{2}& =x\\ {z}^{2}-{x}^{2}& =y\end{array}$

klemmepk

We have
$\begin{array}{rl}{x}^{2}-{y}^{2}& =z\\ {y}^{2}-{z}^{2}& =x\\ {z}^{2}-{x}^{2}& =y\end{array}$
Adding the first two equations, we get that

Similarly, we get that

Hence, there are $8$ possible choices of getting $3$ equations. But the choice

gives no solution. The rest of the seen choices give the following solutions
$\left(0,0,0\right);\left(-1,0,1\right);\left(0,1,-1\right);\left(1,-1,0\right)$

Roland Waters

Sum both sides and get $x+y+z=0$, so you're on a plane.
Further hint:
$z=\left({x}^{2}-{y}^{2}\right)=\left(x-y\right)\left(x+y\right)=-z\left(x-y\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\dots$

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