How can I solve this system (through approximate or exact methods)?

Question

I have a 4 equations 4 unknowns system as i attached it. I want to calculate T1, T2, P1 and P2 through solve this system and i have coded it in mathematica but it could not obtain the answer by solve command. Except this 4 unknowns, all of the other symbols are parameters and Gamma is a parameter equal or greater than 1. Can someone tell me how can i obtain mentioned variables values through any exact or approximate methods? Thank youenter image description here

Answer 1

You can reduce the system of $4$ equations in $4$ unknowns to a system of $2$ equations in $2$ unknowns, either by eliminating $p_1$ and $p_2$ through linear superposition (as they only occur linearly) or by solving ($11$) for $T_1$ and ($12$) for $T_2$ and substituting into ($9$) and ($10$). In either case the resulting equations will be transcendental, and you'll probably need to solve them numerically; but that might be significantly easier than solving the original system.