@vv I agree that the inequalities do count, as I said, the inequalities in this system are used to isolate the solution, choose the positive solutions, to be specific. However, the system f1->f7 defines the curves.
I've used your suggestion, tried to derive f7 and subs it into other equations, I got the new system on x,y,t1 which is good. Thanks for that.
But actually, my goal is to find a general method for this kind of problem. This is a small system; there are systems where I have to deal with a large number of equations. I though there would be the method that reduces the system into a characteristic polynomial. Then we could use some sort of rational parametrization to plot the set easily (the set would be lines, curves, or surfaces).
Anyway, for this general problem, it seems I have to use solve or the tools in DirectSearch package to get the solution.