You forgot to provide the objective function. When you use "minimize" you sure have to tell Prime what you want to be minimized.
As an example I assumed that you want the expression you tried to evaluate at the top of the solve block to be minimized.
So define that expression as a function of the seven variables and use it as the first argument in minimize(f,...).

In case you want all values to be positive, you would have to add this as a constraint:

There still are some negative values in the solution but they are very close to zero. See it as numerical inaccuracies.
In case this is not acceptable, you can "weight" the condition by demanding that, lets say 1000 times the values must be greater than zero

Concerning the algorithm used there is not much documentation but Prime sure does not make use of the simplex method.
Only reference to the algorithms used is a small chapter in the help: https://support.ptc.com/help/mathcad/r9.0/en/#page/PTC_Mathcad_Help/solvers_from_KNITRO.html