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Let \begin{equation} x_1 + x_2 + \dots+ x_n = A \end{equation}

The value of $x_1$ to $x_n$ is not given. Suppose we have $n$ variables $y_1, y_2,\dots, y_n$. Is there any way to find $x_1y_1 + x_2y_2 + x_3y_3 + \dots+ x_ny_n$, based on the value of $A$ and $y_1$ to $y_n$?

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    $\begingroup$ Very unlikely. Any permutation of $x_1, \ldots, x_n$ preserves the identity $x_1 + \ldots + x_n = A$, but may change the value of $x_1 y_1 + \ldots + x_n y_n$ $\endgroup$
    – Martin R
    Dec 26, 2017 at 9:39
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    $\begingroup$ Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. $\endgroup$
    – José Carlos Santos
    Dec 26, 2017 at 9:41
  • $\begingroup$ You have 1 equation for $n$ unknown variables. So in general it cant work. Maybe you missed to mention that it is a polynomial function $f_m(x) = y$ of degree $m$. Then you could solve it for the case of $m=0$. $\endgroup$
    – Raphael J.F. Berger
    Dec 26, 2017 at 9:54
  • $\begingroup$ And likewise for the case of $n=1$ the system is solvable. $\endgroup$
    – Raphael J.F. Berger
    Dec 26, 2017 at 10:02

1 Answer 1

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In most cases no, the answer is yes iff $y_i=y_j\forall i,j\in\{1,2,\cdots,n\}$

Consider $x_1 + x_2 + \dots+ x_n = A$ and that the condition I stated are false.

In this case I can rewrite the $y_1x_1 +y_n x_2 + \dots+ y_nx_n=B$ it as $w_1u_1+w_{2}u_{2}+\cdots+w_\ell u_\ell=B$ where $1<\ell\le n$, $w_k=y_h$ for some $h$ and $u_k$ is combination of some values of $x_s$.

This is equation is the most reduce form, i.e. has the least amount of variables we can get, so unless I have only one variable I can't solve it, but one of my conditions is that not all of $y_i$ are equal, so I have at least 3 variables thus I can't solve this.

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  • $\begingroup$ It does not work even if $A= 0$: $(+1)1 + (-1)2 = -1$, $(-1)1 + (+1)2 =+1$. $\endgroup$
    – Martin R
    Dec 26, 2017 at 12:26
  • $\begingroup$ @MartinR you are right, I made a mistake $\endgroup$
    – ℋolo
    Dec 26, 2017 at 12:34

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