Suppose you're given a set of $n$ functions of $k$ variables\begin{align}f_1(x_1, x_2,\ldots, x_k)& =0\\f_2(x_1, x_2,\ldots, x_k)& =0\\ & {}\ \vdots \\f_n(x_1, x_2,\ldots, x_k)& =0 \end{align}
And you want to eliminate $n-1$ $x$s to give \begin{align}g_1(x_1, x_2,\ldots, x_{k-n-1})& =0 \end{align} The standard way to keep out of trouble would be to eliminate the same variable from one function paired off with each function in turn to give $n-1$ functions of $k-1$ variables. The same procedure is carried on this new set of functions and so on until you end up with the requred function at the end.
What other strategies work, in particular those that may carry some advantages?