Though we proceed through row operations efficiently we should drill down to see is really going on when we perform row operations. We will use the explicit representation to illustrate it. Suppose we want to replace the second row in
\begin{equation*}
\begin{array}{rcrcrcrr}
x \amp + \amp y \amp + \amp 3z \,\amp=\amp5\\
2x \amp - \amp y \amp + \amp 8z \,\amp =\amp-1
\end{array}
\end{equation*}
with two times the first row plus the second, which we may denote by \(2R_1+R_2\rightarrow R_2.\) The process proceeds as follows. First, assume the equalities hold for some \(x,y,z\) and fix such \(x,y,z.\) By ABK 6 multiplying both sides the first equation by \(-2\) preserves equality:
Simplifying the LHS (by ABK 1) and substituting \(-10\) for \(-2x-2y-6z\) on the RHS (by ABK 13, permissible only because we assumed that they were equal!) yields
We will on occasion find it useful to reorder our variables; it is certainly easier to keep track of variables when all pivot variables come before all free variables. Will that be a problem? No.
