left[x_] := -c Exp[alpha x]

box[x_] := a Sin[k x] + b Cos[k x]

right[x_] := c Exp[-alpha x]

dleft[x_] = D[left[x], x]

dbox[x_] = D[box[x], x]

dright[x_] = D[right[x], x]

Eliminate[{

left[-l] == box[-l] &&

dbox[l] == dright[l] &&


   right[l] == box[l] &&

dleft[-l] == dbox[-l]
  }, {a, b, c}, InverseFunctions -> True]

Eliminate[{left[-l] == box[-l] && dbox[l] == dright[l] &&
   right[l] == box[l] && dleft[-l] == dbox[-l]
  }, {a, b}, InverseFunctions -> True]