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I have the following system of linear equations: \begin{align*} \pi_0=\frac{a_o(A_0(\rho))^{\frac{1}{\gamma-1}}-\pi_1\frac{A_1^r-\frac{A_0(r)+A_{01}(r)}{P_0(r)+P_{01}(r)}P_1(r)}{P_0(r)}\left(\frac{A_0(r)P_0(\rho)}{P_0(r)}\right)^{\frac{1}{\gamma-1}}-\pi_2\frac{A_2(r)-\frac{A_0(r)+A_{02}(r)}{P_0(r)+P_{02}(r)}P_2(r)}{P_0(r)}\left(\frac{A_0(r)P_0(\rho)}{P_0(r)}\right)^{\frac{1}{\gamma-1}}}{(A_0(\rho))^{\frac{1}{\gamma-1}}+\frac{A_0(r)}{P_0(r)}\left(\frac{A_0(r)P_0(\rho)}{P_0(r)}\right)^{\frac{1}{\gamma-1}}} \end{align*} $$\pi_1 = \frac{a_1 A_{01} (\rho)^{\frac{1}{\gamma-1}} - 1.53^{\frac{\gamma}{1-\gamma}} \left( P_{00}(\rho) \frac{A_{01} (r) - P_{01}(r) \frac{A_{00}(r) + A_{01}}{P_{00}(r) + P_{01} (r)}}{P_{00}(r)}\right)^{\frac{1}{\gamma -1}} \frac{\pi_0 A_{00}(r) + \pi_2 A_{02} (r) - P_{02} (r) \pi_2 \frac{A_{00}(r) + A_{02}}{P_{00}(r) + P_{02} (r)}}{P_{00}(r)}}{A_{01}(\rho)^{\frac{1}{\gamma-1} } + \left(\frac{A_{00}(r) + A_{01}}{P_{00}(r) + P_{01} (r)} \right)^{\frac{\gamma}{\gamma - 1}} P_{01} (\rho)^{\frac{1}{\gamma-1}} + \left( \frac{A_{01} (r) - P_{01}(r) \frac{A_{00}(r) + A_{01}}{P_{00}(r) + P_{01} (r)}}{P_{00}(r)}\right)^{\frac{\gamma}{\gamma-1}} P_{00}(\rho)^{\frac{1}{\gamma-1}} 1.53^{\frac{\gamma}{1-\gamma}}}$$

$$\pi_2 = \frac{a_2 A_{02} (\rho)^{\frac{1}{\gamma-1}} - 1.53^{\frac{\gamma}{1-\gamma}} \left( P_{00}(\rho) \frac{A_{02} (r) - P_{02}(r) \frac{A_{00}(r) + A_{02}}{P_{00}(r) + P_{02} (r)}}{P_{00}(r)}\right)^{\frac{1}{\gamma -1}} \frac{\pi_0 A_{00}(r) + \pi_1 A_{01} (r) - P_{01} (r) \pi_1 \frac{A_{00}(r) + A_{01}}{P_{00}(r) + P_{01} (r)}}{P_{00}(r)}}{A_{02}(\rho)^{\frac{1}{\gamma-1} } + \left(\frac{A_{00}(r) + A_{02}}{P_{00}(r) + P_{02} (r)} \right)^{\frac{\gamma}{\gamma - 1}} P_{02} (\rho)^{\frac{1}{\gamma-1}} + \left( \frac{A_{02} (r) - P_{02}(r) \frac{A_{00}(r) + A_{02}}{P_{00}(r) + P_{02} (r)}}{P_{00}(r)}\right)^{\frac{\gamma}{\gamma-1}} P_{00}(\rho)^{\frac{1}{\gamma-1}} 1.53^{\frac{\gamma}{1-\gamma}}}$$ Which i want to solve for $\pi_0, \pi_1$ and $\pi_2$ and I know, that it should result in only real solutions.

Where $a, A, P \in \mathbb{R}$ for all indices and $\gamma \in (-\infty, 1) \backslash \{ \varnothing, 0 \}$.

I have 2 problems.

  1. I can't find software, that can for choices of a,A,P solve this. I suspect that it is because $\left( P_{00}(\rho) \frac{A_{01} (r) - P_{01}(r) \frac{A_{00}(r) + A_{01}}{P_{00}(r) + P_{01} (r)}}{P_{00}(r)}\right)^{\frac{1}{\gamma -1}}$ is complex and R (the language im using) can't cancel it out with other terms.
  2. Do you know any software, that is good at analytically solving such expressions. I would like to have a solution, that I can work towards. I already tried Maple without any luck.
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