Hello guys i'm trying to derive a general formula for something and this is the system of equations i get. I'm hoping to get a simple and beautiful solutions but i fail. The solution i got my elimination was
$$q-l=a(p^2-k^2)+b(p-k)$$
but this yields an answer for $a+b$ and i want to get a solution for only $a$ and only $b$. I tried to use substitution but the answer yielded was really long and ugly. Could you please look over this and write whether it has good answers or not. Here's the system:
$$q=ap^2-am^2+bp-bm+n$$ $$l=ak^2-am^2+bk-bm+n$$
Try solving for $a$ in terms of $b$ in the first equation, and then plug that into the second equation. For example:
$q = a(p^2-m^2)+b(p-m)+n$, so $a(p^2-m^2)=q - b(p-m) - n$, and $a = \frac{q-b(p-m)-n}{p^2-m^2}$.
Plug $a$ into the second equation now.
$l = a(k^2-m^2)+b(k-m)+n=\frac{q-b(p-m)-n}{p^2-m^2}(k^2-m^2)+b(k-m)+n$
Now you have $b(k-m) = l-n-\frac{q-b(p-m)-n}{p^2-m^2}(k^2-m^2)$, and $b = l-n-\frac{q-b(p-m)-n}{(p^2-m^2)(k-m)}(k^2-m^2)$
There are simplifications you can do along the way, I'm just doing the straight forward approach. Plug $b$ into the equation for $a$ and you'll get some mess that you should be able to simplify.