I have this system of 2 equations in 4 unknowns:
$$
\begin{cases}
\displaystyle y_1=\pi_1 \frac{1}{1+e^{\alpha_1}}+\pi_2 \frac{1}{1+e^{\alpha_2}}+\pi_3 \frac{1}{1+e^{\alpha_3}}+\pi_4 \frac{1}{1+e^{\alpha_4}}\\
\displaystyle y_2=\pi_1 \frac{e^{\alpha_1}}{1+e^{\alpha_1}}+\pi_2 \frac{e^{\alpha_2}}{1+e^{\alpha_2}}+\pi_3 \frac{e^{\alpha_3}}{1+e^{\alpha_3}}+\pi_4 \frac{e^{\alpha_4}}{1+e^{\alpha_4}}\\
\end{cases}
$$
The unknowns are $\alpha_1,\alpha_2,\alpha_3,\alpha_4$; all other terms are known.
I have never taken any course on solving systems of equations and my questions are very basics (apologies in advance): what can I say about the number of solutions? Are there infinite solutions and why? Is there any way to pin down a unique solution?
Best Answer
Adding together the two equations and collecting terms gives $$y_1+y_2=\sum_{i=1}^4\frac{\pi_i}{1+e^{\alpha_i}}(1+e^{\alpha_i})=\sum_{i=1}^4{\pi_i}.$$
If false, there are no solutions. If true, you have one equation with four unknowns, which is underdetermined and this has an infinite number of solutions over the reals.