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Say you have a solution $\textbf{x}=(x_1,x_2,\ldots,x_n)$ to a system of equations. It turns out that $-\textbf{x}$ is also a solution. Is there accepted terminology for such a pair of solutions? (I don't think conjugate is it.) Is there terminology for the system that admits such solutions? (I don't think symmetric is general enough.)

Assume we're dealing with real numbers here.

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I don't think there is. Were I in that position, I'd just say that system so-and-so has solutions $\mathbf x=(\pm x_1,\dots,\pm x_n)$. –  J. M. Oct 24 '11 at 14:59
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@J.M., your notation implies $(x_1,-x_2,x_3,...)$ is a solution, which is not what OP has written. –  Thomas Andrews Oct 24 '11 at 15:12
    
My initial instinct was to call them "antipodal," but that only really works if the points are confined to a sphere. Yes, implicitly $\textbf{x}$ and $-\textbf{x}$ are on the same sphere with origin $0$, and they are antipodal on that sphere, but I think that's a stretch of the term. –  Thomas Andrews Oct 24 '11 at 15:15
    
@Thomas: probably with the qualifier "with same upper signs or lower signs..." then. –  J. M. Oct 24 '11 at 15:18
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@J.M.: Not clear why you wouldn't just write that as $\pm \textbf{x}$ –  Thomas Andrews Oct 24 '11 at 15:29

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Standard in calculus would be "symmetric with respect to the origin" or "symmetric by reflection in the origin" or some variant.

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