Linear system/Equating method/Remark

Another method to solve a linear system is the equating method. Here, the variables are also successively eliminated, but in another way. In this method, in every equation ${}G_{i}$, ${}i=1,\ldots ,m$, we isolate one fixed variable, say ${}x_{1}$. Suppose that (after reordering) ${}G_{1},\ldots ,G_{k}$ are the equations where the variable ${}x_{1}$ occurs with a coefficient different from ${}0$. These equations are brought into the form

$G_{i}':\,\,\,x_{1}=F_{i},$

where in ${}F_{i}$, the variable ${}x_{1}$ does not occur. The linear system consisting in

$G_{1}',F_{1}=F_{2},F_{1}=F_{3},\ldots ,F_{1}=F_{k},G_{k+1},\ldots ,G_{m}$

is equivalent to the original system. We continue with this system without ${}G_{1}'$.