Yes joigus,
I think we start to be on the right way.
So, as you already understood, my system becomes:
\[ \begin{cases} −x^2 + 10x − 1/2y^2 + 6y − K = 0 \\ −2x^2 + 20x + 3/2y^2 − 18y = 0 \\ (−2x+10)−(−4x+20)=0 \end{cases} \]
And like you said, from the 3rd equation I get \(x=5\). Substituting this in the 2nd equation, we get \(y=4.367\) or \(y=7.633\). Finally substituting x and y in the 1st equation, we get \(k=41.667=\frac{125}{3} \) .
So for this example imposing the 3rd equation as the difference of the derivates respect x, works.
You can try to do the same with the second example, but in this case you have to derivate respect y to obtain the 3rd equation. Solving the system, you will get:
\[x=1.6626 \\ y=6.0 \\ k=45.0\]
that is exactly again the maximum value of k that we were looking for.
The reason is that the 3rd equation adds the condition of tangency between the 2 intersection of the surfaces represented by the first 2 equations with the plane xy. So , because they are tangent, the solution of the system that is the intersections of the previous curves, can only be 1. I show you both cases in the following picture:
So, they case on the left that has 2 solutions is for \(k<125/3\) and the case on the right with just 1 solution (tangency between the 2 curves is \(k=125/3\). If we have \(k>125/3\), the 2 curves doesn't intersect and there isn't any solution.