I try to solve for variable P3. Sage gives me an answer for P3 but it still contains P3 on the right hand side?
sage: A1, A2, A3, P1, P2, P3, u1, u2, u3, r1, r2, r3, g, C, M1, M2, M3 = var('A1 A2 A3 P1 P2 P3 u1 u2 u3 r1 r2 r3 g C M1 M2 M3') sage: eq1=A3(P1-P3)==A3r3u3^2-(A2r2u2^2+A1r1u1^2) sage: eq5a=solve([eq2],r3) sage: eq5b=u3^2+2g/(g-1)(P3/(eq5a[0].rhs()))-2C==0 sage: eq6=solve([eq5b],u3) sage: eq7a=eq1/A3 sage: eq7b=eq7a.subs(-A3r3u3^2 == -(A2u2r2+A1r1u1)u3) sage: eq7b P1 - P3 == -(A1r1u1^2 + A2r2u2^2 + (A3P3g + sqrt(A3^2P3^2g^2 + 2((g^2 - 2g + 1)A1^2r1^2u1^2 + 2(g^2 - 2g + 1)A1A2r1r2u1u2 + (g^2 - 2g + 1)A2^2r2^2u2^2)C))(A1r1u1 + A2r2u2)/(A1(g - 1)r1u1 + A2(g - 1)r2u2))/A3
sage: eq7c=eq7b.subs(u3==eq6[0].rhs()) #this equation should have multiple answers+_, but it does not have multiple sage: eq7=solve([eq7c],P3)
sage: eq7 [P3 == (A1r1u1^2 + A2r2u2^2 + A3P1 - (A1r1u1^2 + A2r2u2^2 + A3P1)g - sqrt(A3^2P3^2g^2 + 2((g^2 - 2g + 1)A1^2r1^2u1^2 + 2(g^2 - 2g + 1)A1A2r1r2u1u2 + (g^2 - 2g + 1)A2^2r2^2u2^2)*C))/A3]