How would one solve algebraically the following system of algebraic equations? 
$$f(a,b):=a(1-b)+ab\frac a{a+b}.$$
$$u = f(a,b),\quad v = f(b,a).$$
Solve algebraically $(a,b)$ in terms of $(u,v)$




---


Multiplying both sides of the equations by $a+b$ would give us a system of cubic equations. But that does not seem to help much in solving the equations.



Algebraic solution for a system of algebraic equations?
This is mechanized in current CASes, e.g. the command of Maple 2019.1



```
solve({a*(1 - b) + a*b*a/(a + b) = u, eval(a*(1 - b) + a*b*a/(a + b), {a = b, b = a}) = v}, {a, b}, explicit);

```

performs a long output which can be seen [here](https://www.dropbox.com/s/wd89oddvbjt99bk/solution.pdf?dl=0) exported as a PDF file.


Addition. The command of Mathematica



```
Reduce[{a*(1 - b) + a*b*a/(a + b) == u, b*(1 - a) + b^2*a/(a + b) == v}, {a, b}]//ToRadicals

```

produces



> 
> (u == 1 - v && 
>  a == 1 && (b == 1/2 (v - Sqrt[v] Sqrt[4 + v]) || 
>  b == 1/2 (v + Sqrt[v] Sqrt[4 + v]))) || ((a == (2 + u)/4 - 
>  1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 
>  1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))) - 
>  1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - 
>  v - (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(
>  1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 
>  8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
>  v + 1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 
>  5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))))) || 
>  a == (2 + u)/4 - 
>  1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 
>  1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))) + 
>  1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - 
>  v - (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(
>  1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 
>  8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
>  v + 1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 
>  5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))))) || 
>  a == (2 + u)/4 + 
>  1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 
>  1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))) - 
>  1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - 
>  v - (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(
>  1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 
>  8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
>  v + 1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 
>  5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))))) || 
>  a == (2 + u)/4 + 
>  1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 
>  1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3))) + 
>  1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - 
>  v - (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 
>  3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(
>  1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 
>  8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
>  v + 1/3 (2 u + v) + (2^(
>  1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 
>  5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
>  3 2^(1/3)))((2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - 
>  v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 
>  3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 
>  27 (-2 - u)^2 (-u^2 - u v) - 
>  72 (2 u + v) (-u^2 - u v) - 
>  9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 
>  27 (u + v - u v - v^2)^2)^2))^(1/3)))))) && -1 + 
>  a != 0 && b == (a - u - v)/(-1 + a) && a^2 - u - v != 0)
> 
> 
> 


