<s>[INST] Find the sum of all values of $x$ which satisfy the following systems of equations: Systems of equations are as follows:
$$\left\lbrace\begin{aligned}
y&=x^2-5x+5\\
z&=x^2-12x+35\\
y^z&=1\\
x,&y,z \in\mathbb{R}\\
\end{aligned}\right.$$
I got the obvious $x=5,7,1$ and $4$ but apparently there are other answers as the sum should equal to $20$ and the answers I got sum to $17$; and if $z=0$, $y$ can be anything and if $y=1$, $z$ can be anything.
 [/INST] There are infinitely many values of $x$ satisfying this system. For if you write the second equation as $z=x^2-5x+5-7x+30,$ and substitute for $x^2-5x+5$ using the first equation, we get $$z=y-7x+30.$$ Finally, the last equation implies $z=0$ or $y=1.$
If $z=0,$ then $y=7x-30$ and if $y=1,$ then $z=31-7x.$ Thus we have the families of solutions $(\lambda,7\lambda-30,0)$ and $(\mu, 1, 31-7\mu).$ The last case is when $z=0$ and $y=1,$ which gives the singleton $(31/7, 1, 0).$ Thus, you literally can't add all the values of $x$ satisfying the relation as $\lambda$ and $\mu$ can be any real number.
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