Solving the following equation for $y_\infty$ took hours with me, all ways were not feasible. I tried WolframAlpha, it says the variable is not present in the equation.
$x_1,y_1,x_2,y_2,x_3,$ and $y_3$ are all known values.
$y_\infty>y_1$, $y_\infty>y_2$.
$x_1 \ne x_2$.
How to solve for $y_\infty$?
Any help would be appreciated. THANKS!

It would have been a good idea you give the link to the corresponding Wolfram Alpha page.
On the other side, I think that you will never get an explicit solution and you need numerical methods.
Could you provide a set of values for $(x_1,y_1,x_2,y_2,x_3,y_3)$ ? I should look at the problem.