Computational methods for manipulating sets of polynomial equations are becoming of greater importance due to the use of polynomial equations in various applications. In some cases we need to eliminate variables from a given system of polynomial equations to obtain a "symbolically smaller" system, while in others we desire to compute the numerical solutions of non-linear polynomial equations. Recently, the techniques of GrΓΆbner bases and polynomial continuation have received much attention as algorithmic methods for these symbolic and numeric applications. When it comes to practice, these methods are slow and not effective for a variety of reasons. In this paper we present efficient techniques for computing multipolynomial resultant algorithms and show their effectiveness for manipulating system of polynomial equations. In particular, we present efficient algorithms for computing the resultant of a system of polynomial equations (whose coefficients may be symbolic variables). The algorithm can also be used for interpolating polynomials from their values and expanding symbolic determinants. Furthermore, it is possible to come up with tight bounds on the running time and storage requirements of the algorithm. Finally, we use the symbolic elimination algorithm to compute the real or complex solutions of non-linear polynomial equations. It reduces the problem to finding roots of univariate polynomials. We also discuss the implementation of these algorithms and discuss their performance on some applications.