In this paper we present a solution to the inverse kinematics problem for serial manipulators of general geometry. The method is presented in detail as it applies to a 6R manipulator of general geometry. The equations used are derived using a linearization method and dialytic elimination. In doing this, all variables except one, a tangent half angle of a joint variable, can be eliminated. The result is a 16 by 16 matrix in which all terms are linear in the suppressed variable. The unique design of this matrix allows the suppressed variable to be solved as an eigenvalue problem. Substituting these values of the suppressed variable back into the equations, all other joint variables can be found using linear equations. The result is the 16 solutions expected for the 6R case. The same technique is also applicable to manipulators with prismatic joints. We present the solution technique for all six possible 5R,P manipulators through numerical examples. The primary distinction between the technique presented in this paper and recently published Raghavan and Roth (90a,b.c) solution is that they removed two known spurious imaginary roots of multiplicity four to obtain a 16th order polynomial for 6R and 5R,P cases. In our formulation, the 16th degree polynomial can be derived directly without having to remove any spurious imaginary roots. Another distinction is that the solution procedure presented in this paper can be reduced to an eigenvalue problem. This results in significant gains in computation time.