Is it possible to have three real numbers that have both their sum and product equal to $1$?

I have to solve $ x+y+z=1$ and $xyz=1$ for a set of $(x, y, z)$. Are there any such real numbers?

Edit : What if $x+y+z=xyz=r$, $r$ being an arbitrary real number. Will it still be possible to find real $x$, $y$, $z$?

Answer

There are infinite solutions. $z=1-x-y$ is a plane’s equation and $z=\dfrac{1}{xy}$ is a complicated curve, but their intersection presents infinitely many real points.

Intersection

At first I would try some trivial values like $x=0$ or $x=1$ or $x=-1$ and check which of them works and which doesn’t.

For instance, if $x=0$, then $xyz=0$ doesn’t work.

If $x=1$, $x+y+z=1\rightarrow y+z=0\rightarrow z=-y$ means that $xyz=1\rightarrow y^{2}=-1$ is a complex number also doesn’t work.

Then $x=-1$ is a good attempt because $y+z=2$ and $yz=-1$ looks promising.
\begin{equation*}
yz=-1
\end{equation*}
\begin{equation*}
z=-\frac{1}{y}
\end{equation*}
then,
\begin{equation*}
y+z=2
\end{equation*}
\begin{equation*}
y-\frac{1}{y}=2
\end{equation*}
\begin{equation*}
y^{2}-1=2y
\end{equation*}
\begin{equation*}
y^{2}-2y-1=0
\end{equation*}
\begin{equation*}
\Delta=4+4=8
\end{equation*}
\begin{equation*}
y=\frac{2\pm2\sqrt{2}}{2}=1\pm\sqrt{2}
\end{equation*}
\begin{equation*}
\therefore z=1\mp\sqrt{2}
\end{equation*}

Thus, both $\left(-1,1-\sqrt{2},1+\sqrt{2}\right)$ and $\left(-1,1+\sqrt{2},1-\sqrt{2}\right)$ are valid answers.

I could not comment above for the lack of space. Here are some notes about their signals:

As noticed before none of the variables can be zero, because $xyz\neq0$.

Also the three variables cannot be all positive, because if $x>0$, $y>0$, $z>0$, then it would put them under the interval $0<x,y,z<1$, if one of them would be greater than $1$, for example, $x>1$ then the sum $x+y+z$ would be greater than $1$, since $x>1$, $y>0$, $ z>0$. The problem of that interval is $0<y<1$ would mean that $\dfrac{1}{y}>1$ and for the same reason $0<z<1$ would mean $\dfrac{1}{z}>1$. But $xyz=1$ would make $x=\dfrac{1}{y}\cdot\dfrac{1}{z}>1$ and that would be impossible, so the three variables cannot be all positive and they cannot be localized on that interval.

On the other hand, you cannot have just one negative variable or three negative variables because their product would be negative, so you must have exactly two negative and one positive variables.

Attribution
Source : Link , Question Author : bikalpa , Answer Author : J. M. ain’t a mathematician

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