Thank you for your suggestions. They gave me some idea but there is still one thing that I can't figure out. In the examples you gave the values of k are entered explicitly by hand. But in my case the parameter k (in fact, not just one k but 4 different parameters k1, k2, k3, k4) will be read from file. And then I will need to solve the system of equations for each combination of the values of these 4 parameters. That will be done hundreds of times. In that case, according to my understanding, I will need to have a cycle, in each step of the cycle k1...k4 will be defined but in the expression to be solved they must be entered as symbols then, not as fixed numbers. How can that be done?

You can use delayed evaluation:

sol[k_] := FindRoot[Sin[x] + Exp[x] == k, {x, 0}];
sol[0]
sol[1]

This way the numerical algorithms of FindRoot will be called only after a value has been decided for k.