Two non-linear simultaneous equations

Question

what is wrong with this? It runs forever and never displays anything on the screen.

NSolve[{E^-k == (Cg + m)/(Cs - (Cs/q) + ((E^(\[Theta]q\[Alpha]E^(-k))       
-1)/(q\[Theta]E^(-k) \[Tau] (E^(\[Theta]q\[Alpha]E^(-k)) + \
\[Tau] - 1)))) \[Tau] + (\[Alpha]/\[Tau]) + m,
(E^(\[Theta]q\[Alpha]E^-k) - 1)/(\[Tau] + 
 E^(\[Theta]q\[Alpha]E^(-k)) -1) == \[Theta]E^(-k) \[Tau] (Cs + (q\[Beta]/(1 
 -E^(-k) \[Tau])^2)) /. {q -> 0.5, \[Alpha] -> 4, Cg -> 5, 
 m -> 5, Cs -> 7, \[Beta] -> 0.5, \[Theta] -> 0.3}}, {\[Tau], k}, Reals]

Answer 1

FindRoot[{
  E^-k == (Cg+m)/(Cs-Cs/q+(E^(θ*q*α*E^-k)-1)/(q*θ*E^-k*τ*(E^(θ*q*α*E^-k)+τ-1)))*τ+α/τ+m,
  (E^(θ*q*α*E^-k)-1)/(τ+E^(θ*q*α*E^-k)-1) == θ*E^-k*τ*(Cs+q*β/(1-E^-k*τ)^2)} /.
{q->1/2, α->4, Cg->5, m->5, Cs->7, β->1/2, θ->3/10}, {{τ, 1}, {k, 1}}, 
WorkingPrecision -> 16]

immediately returns

{τ->4.045735266605817, k->37.94584124554267}

but if you increase that working precision you get different values for k from 19 out to 106 and beyond so there may be multiple solutions for this.

If for each of your equations you Plot the left hand side-right hand side with that value of τ substituted in and for values of k ranging from 1 to 106 then you can see the somewhat inconvenient behaviour of your two equations.