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 $\gamma_1,\ldots,\gamma_N$?