Its a very simple problem to a homework question that I can easily solve, but if time-willing I like to go about it a more elegant way.
So lets say you need to create a set of 5 numbers whose median is m, and whose mean is n. I have specific values, but I'm looking for a general way to do this.
I'm trying to accomplish this by putting it into matrix form and use linear algebra to solve this. Is it possible?
Answer:
median 10, mean 7, 5 numbers. I left 10 out because it was known. And took away from the elegance.
$ \\ \begin{bmatrix} 2 & 1 & 11 & 11 \\ 1 & 2 & 11 & 11 \\ 1 & 1 & 12 & 11 \\ 1 & 1 & 11 & 12 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}$
$ 7 = \frac{x_1 + x_2 + 10 + x_3 + x_4}{5} \\ 35 = x_1 + x_2 + 10 + x_3 + x_4 \\ 25 = x_1 + x_2 + x_3 + x_4 \\ 2 \leq x_1+x_2 \leq 3 \\ 22 \leq x_3 + x_4 \leq 23 $
An approach could be linear programming. $A x \leq b$
Typical $A$ matrix would be defined by constraints such as $x_1 \leq x_2 \leq x_3 = m \leq x_4 \leq x_5$ and $ \sum x_i / 5 = n$
There are some methods to convert such requirements to elements of $A$, I hope you know them, because it is a bit too much to explain. E.q. $x_1 \leq x_2 $ would be $x_1 - x_2 \leq 0$ and as such a row in $A$ of [1 -1 0 0 0] and an element in $b$ of 0.