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If $y=-x$ and $\displaystyle \frac{y}{x-z}=\frac{x}{y}$ then either $x:y:z=1:-1:0$ or $x:y:z=-1:+1:0$.

Is this correct? If not why?

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Looks good to me. Btw, there is no difference between the two answers you gave. – soandos Jun 4 '11 at 20:25
up vote 1 down vote accepted

If $z=x$ then $x-z = 0$ and then the LHS is undefined.

Since $x=-y$ the RHS is equal to $-1$, therefore $\frac{y}{x-z} = -1$, which means $y = z-x$ and so $z=0$.

Note, however, that $x,y$ can be pretty much anything under $z=0$, not just $\pm 1$.

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I believe OP is saying the ratio of $x$ to $y$ must be -1, which is correct. – Ross Millikan Jun 4 '11 at 23:06
@Ross: I have not used (or even seen) the $x\colon y\colon z$ notation for years now, but when you say it that way it comes back to me, and it seems that you are right. :-) – Asaf Karagila Jun 5 '11 at 7:39

So long x $\neq$ z and y$\neq$ 0. This should be stated along with the derived conclusion about the ratios.

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