from sympy.solvers import solve
from sympy import *
P, K, L, PL, Pt, Lt= symbols('P, K, L, PL, Pt, Lt')
solve([K-(P * L)/PL, Pt- P - PL, Lt- L - PL], P,L,PL)
[(-K/2 - Lt/2 + Pt/2 - sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2,
-K/2 + Lt/2 - Pt/2 - sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2,
K/2 + Lt/2 + Pt/2 + sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2),
(-K/2 - Lt/2 + Pt/2 + sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2,
-K/2 + Lt/2 - Pt/2 + sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2,
K/2 + Lt/2 + Pt/2 - sqrt(K**2 + 2*K*Lt + 2*K*Pt + Lt**2 - 2*Lt*Pt +
Pt**2)/2)]
The 1st solution is different. the solution in sage seems wrong.
sage:
var('P, K, L, PL, Pt, Lt')
solve([K-(P * L)/PL==0, Pt- P - PL==0, Lt- L - PL==0], P,L,PL)
[[P == 2*K*Pt/(K + Lt - Pt + sqrt(K^2 + 2*K*Lt + Lt^2 + 2*K*Pt - 2*Lt*Pt
+ Pt^2)), L == -1/2*K + 1/2*Lt - 1/2*Pt + 1/2*sqrt(K^2 + 2*K*(Lt + Pt) +
Lt^2 - 2*Lt*Pt + Pt^2), PL == 1/2*K + 1/2*Lt + 1/2*Pt - 1/2*sqrt(K^2 +
2*K*Lt + Lt^2 + 2*K*Pt - 2*Lt*Pt + Pt^2)], [P == 2*K*Pt/(K + Lt - Pt -
sqrt(K^2 + 2*K*Lt + Lt^2 + 2*K*Pt - 2*Lt*Pt + Pt^2)), L == -1/2*K +
1/2*Lt - 1/2*Pt - 1/2*sqrt(K^2 + 2*K*(Lt + Pt) + Lt^2 - 2*Lt*Pt + Pt^2),
PL == 1/2*K + 1/2*Lt + 1/2*Pt + 1/2*sqrt(K^2 + 2*K*Lt + Lt^2 + 2*K*Pt -
2*Lt*Pt + Pt^2)]]
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