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This applies to the factored system,

LaTeX: E_1 E_2 ... E_n x = b,

where each LaTeX: E_i is an elementary matrix. The recursion is:

E_1 x_1 & = & b \\
E_2 x_2 & = & x_1 \\
& \vdots & \\ 
E_n x_n & = & x_{n-1}

In the end, LaTeX: x = x_n is the solution. Each elementary system is solved as follows. For notational convenience, suppose the system is LaTeX: Ex = y, and LaTeX: v is the distinguished LaTeX: p-th column of LaTeX: E. Then,

LaTeX: x(p) = y(p)/v(p), and LaTeX: x(i) = y(i) - v(i)x(p) for LaTeX: i \ne p.

This is what is done in the (revised) simplex method, and each elementary solution is a pivot operation.

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