Method to see why systems of equations has no solution

I am trying to solve a set of equations two variables as below. Mathematica said that it has no solution. I would like to research more why it has no solution. I tried to use Trace function but it doesn't reveal anything. Is there any way to see why there is no solution for this system of equations?

Solve[{-((3*(-1 + D1 + D2 + D3)*(5*(-1 + D1 + D2) + 7*D3))/(5*(-1 + D1 + D2)^2 +
10*(-1 + D1 + D2)*D3 + 7*D3^2)) == 2,
-((3*D3*(5*(-1 + D1 + D2) + 7*D3))/(5*(-1 + D1 + D2)^2 + 10*(-1 + D1 + D2)*D3 + 7*D3^2)) == 5},
{D1, D2}]

• The system forces D3==0. Solve returns generic solutions and that is not a generic condition. You might instead do: Solve[Numerator[polys] == 0, {D1, D2}, MaxExtraConditions -> 1] to allow for a parameter condition. Commented Apr 26, 2018 at 14:30

There is no solution, because Numerator and Denominator have the same roots.

Bring right side of equations to left side and generate a common denominator

exp2 = eqs[[All, 1]] - eqs[[All, 2]] // Together

(*   {(-25 + 50 D1 - 25 D1^2 + 50 D2 - 50 D1 D2 - 25 D2^2 + 56 D3 -
56 D1 D3 - 56 D2 D3 - 35 D3^2)/(5 - 10 D1 + 5 D1^2 - 10 D2 +
10 D1 D2 + 5 D2^2 - 10 D3 + 10 D1 D3 + 10 D2 D3 + 7 D3^2), (-25 +
50 D1 - 25 D1^2 + 50 D2 - 50 D1 D2 - 25 D2^2 + 65 D3 - 65 D1 D3 -
65 D2 D3 - 56 D3^2)/(5 - 10 D1 + 5 D1^2 - 10 D2 + 10 D1 D2 +
5 D2^2 - 10 D3 + 10 D1 D3 + 10 D2 D3 + 7 D3^2)}   *)


Denominators of both expressions are the same

Denominator[exp2[[1]]] == Denominator[exp2[[2]]]

(*   True   *)


Since denominators shall not go to infinity, you only get solutions, if nominators are zero.

sol = Solve[{Numerator[exp2[[1]]] == 0, Numerator[exp2[[2]]] == 0}]

(*   {{D1 -> 1 - D2, D3 -> 0}}   *)


But unfortunately with this solutions, denominators are also zero.

Denominator[exp2] /. First@sol // Simplify

(*   {0, 0}   *)


And in the limit, you don't get zero.

Limit[exp2, D1 -> 1 - D2]

(*   {-5, -8}   *)

Limit[exp2, D3 -> 0]

(*   {-5, -5}   *)


That means, you can never get a solution.