# Scaling sum of numbers to add up to particular value

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 $\gamma_1,\ldots,\gamma_N$?

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.
• As long as $A_N\neq0$, anyway. – Cameron Buie Nov 17 '12 at 17:18
• @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? – Nicholas Kinar Nov 17 '12 at 17:37