# Merging multiple non linear equations into one

I have 3 nonlinear equations in 3 unknowns,

$$\Large A=\frac{P_L}{1-c}\quad B=\frac{P_L}{1-ce^{-10k}}\quad C=\frac{P_L}{1-ce^{-20k}}$$

where $$A$$, $$B$$, $$C$$ are known values and $$P_L$$, $$c$$ and $$k$$ are unknowns. I want to get them into one single equation in one of three unknowns ($$P_L$$, $$c$$ or $$k$$, does not matter) so I can use one of the root finding methods to determine that single unknown.

• Put $x = e^{-10k}$ then between the first two equations you can show $c=\frac{(B-A)}{(Bx-A)}$. Use the second and third equations to write $\frac{B}{C}$ in terms of c and x. Now sub in your expression for c and multiply out to get a quadratic in x.
– Paul
Dec 19 '20 at 21:32

You have $$c=1-\frac{P}{A}$$
$$ce^{-10k}=1-\frac{P}{B}$$ $$ce^{-20k}=1-\frac{P}{C}$$
Therefore $$(1-\frac{P}{B})^2=(1-\frac{P}{A})(1-\frac{P}{C})$$