Thank you so much Professor Gianluca! You've been so helpful!!!

Are you trying to find solutions where ix==iy?

Thanks Professor Gianluca, I tried to replicate these same steps to do the opposite to plot iy as a function of ix, but how would I then combine them? I'm trying to figure out the nash equilibrium solution for iy and ix, and in easier example, I'd normally be plotting iy as a function of ix, and ix as a function of iy and seeing where they cross, but since they're not showing up in the plot, how would I figure how where these functions intersect?

Thank you again!

f[iy] is a ConditionalExpression. The existence condition is its last element. It seems that there are indeed positive solutions:

In[36]:= Last[f[iy]] // N
f[400.]

Out[36]= iy > 338.568

Out[37]= 0.742669

Thanks for the explanation Gianluca! Plotting it out, it seems like the solution is in the 4th quadrant. In my model, ix and iy can't be negative numbers, I modified the code you had to include the constraints for ix and iy, and the plot turned out empty. I'm not sure if this is because I didn't put the constraint in the right area of the code, or does a positive solution just not exist for ix and iy? (and how would I be able to tell for the latter?) Thank you so much!

f[iy_] = ix /. 
   First@Solve[
     Simplify[(-r + 
           9/40 b ((7 Sqrt[c] Sqrt[r] + 
                  Sqrt[q + b iy q + 49 c r])^2/(1 + 
                  b ix)^2 + (2 q^2)/(Sqrt[
                  q + b ix q + 49 c r] (-14 Sqrt[c] Sqrt[r] + 
                   Sqrt[q + b ix q + 49 c r] + 
                   Sqrt[q + b iy q + 49 c r]))) == 0 && ix >= 0 && 
         iy >= 0) /. {q -> 50, r -> 50, c -> 1/100, b -> 1/10}], ix, 
     Reals];
Plot[f[iy], {iy, -15, 20}]

Hi Gianluca, thanks for giving an alternative way of doing it. I entered that and got this whole blob. I'm not quite sure what kind of "answer" this means. Is this a real number? or is it now a complex number? I'm still new to mathematica so when the output looks like this I'm a bit confused. Thank you!

This variation gives a quick answer:

Solve[Simplify[(-r + 
      9/40 b ((7 Sqrt[c] Sqrt[r] + Sqrt[q + b iy q + 49 c r])^2/(1 + 
           b ix)^2 + (2 q^2)/(
         Sqrt[q + b ix q + 
           49 c r] (-14 Sqrt[c] Sqrt[r] + Sqrt[q + b ix q + 49 c r] + 
            Sqrt[q + b iy q + 49 c r]))) == 0) /. {q -> 50, r -> 50, 
    c -> 1/100, b -> 1/10}], ix, Reals]