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I have a sum of real numbers $A_1 + A_2 + \cdots + A_N = A$ that add up to a known number $A$. All of the $A_1,\ldots,A_N$ are known as well.

Is there a way of scaling the $A_1,\ldots,A_N$ so that the sum of the numbers add up to another known real number $B$ instead of $A$?

So what I am searching for are $\gamma_1,\ldots,\gamma_N$ such that:

$$A_1\gamma_1 + A_2\gamma_2 + \cdots + A_N\gamma_N = B$$

What assumption can I make to ensure uniqueness of the $\gamma_1,\ldots,\gamma_N$? Is it possible to find $\gamma_1,\ldots,\gamma_N$?

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You can make all the $\gamma$'s$= \frac BA$. There are many other solutions-you can make the first $N-1$ anything you want and you will have a linear equation for the last one.

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    $\begingroup$ As long as $A_N\neq0$, anyway. $\endgroup$ – Cameron Buie Nov 17 '12 at 17:18
  • $\begingroup$ @RossMillikan: Thanks, Ross; I really like the $\gamma = \frac BA$ solution. What are the conceptual implications of making $\gamma = \frac BA$? What type of scaling is this? $\endgroup$ – Nicholas Kinar Nov 17 '12 at 17:37
  • $\begingroup$ It is just a linear scaling. I'm not sure what you mean by conceptual implications. It is like you graded on a scale of 0 to 5 and now want 0 to 100. You multiply everything by 20. $\endgroup$ – Ross Millikan Nov 17 '12 at 17:46
  • $\begingroup$ @RossMillikan: Thanks, Ross. That's exactly what I was looking for. $\endgroup$ – Nicholas Kinar Nov 17 '12 at 17:52

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