How to solve this system of equations?

Questions
1

Hi!

I am trying to solve the system of equations in the attached file. What I am looking for are the values of tP and tC, both in (0,1), which satisfy:

exp3= tP/(1 - tP) and exp6 = tC/(1 - tC)

I am using FindRoot, but I have reasons to believe that it is not returning all the solutions. How would you solve this system of equations in Mathematica? Thanks a lot for your help.

Laurent

2

FindRoot only gives one solution for a given starting point. If you want more solutions, you should try more starting points.

3

I have some doubts that there are more than one solution, at least in the unit square. The following plots show ony one likely candidate:

Plot3D[{Sqrt@Norm[{exp3 - tP/(1 - tP), exp6 - tC/(1 - tC)}], 0}, {tC, 
  0, 1}, {tP, 0, 1}, PlotRange -> {-.1, 1}, AxesLabel -> Automatic]
Plot3D[{Sqrt@Norm[{exp3 - tP/(1 - tP), exp6 - tC/(1 - tC)}], 
  0}, {tC, .8, .9}, {tP, 0.5, .9}, PlotRange -> {-.1, 1}]