Solving an exponential multivariate equation

Question

I have the following equation:

p = 1/(1 + E^(-51.179 + 0.849 x1 + 2.3649 x2 + 1.155 x3 + 1.46 x4 + 1.31 x5))

and I want to find the set of values for x1 to x5 such that the fraction be equal to 0.5.In other words, I want x1-x5 values for which p==0.5. I tried NSolve and Reduce but those didn't work. All x1-x5 are between 0 and 10.

I searched the list but I couldn't come up with a solution. Any help is appreciated.

Answer 1

FindInstance[p == 0.5, {x1, x2, x3, x4, x5}]

Gives:

{{x1 -> -3.5 + 4.9 I, x2 -> -0.6 - 1.1 I, x3 -> -3.2 + 0.5 I, 
  x4 -> -0.2 - 2.5 I, x5 -> 45.4637 + 44.3225 I}}

You can also limit the domain and ask to find n instances

   sol = FindInstance[p == 0.5, {x1, x2, x3, x4, x5}, Reals, 3]

{{x1 -> -7.5, x2 -> -3.7, x3 -> -0.5, x4 -> 14.4, x5 -> 35.0001},
 {x1 -> 6.6, x2 -> -11.2, x3 -> -5., x4 -> 5.9,  x5 -> 52.8424}, 
 {x1 -> 11.4, x2 -> 14.9, x3 -> 5.2, x4 -> 11.3, x5 -> -12.3974}}

Check

p /. sol    

{0.5, 0.5, 0.5}

To specifically select values that meet your criteria do the following:

FindInstance[{p == 0.5, x1 > 0 && x1 <= 10, x2 > 0 && x2 <= 10, 
  x3 > 0 && x3 <= 10, x4 > 0 && x4 <= 10, x5 > 0 && x5 <= 10}, {x1, 
  x2, x3, x4, x5}, Reals]

Gives

{{x1 -> 1.06349, x2 -> 4.66239, x3 -> 10., x4 -> 10., x5 -> 10.}}