Linear system/Substitution method/Remark

Another method to solve a linear system is the substitution method. Here, the variables are also successively eliminated, but in another way. If we want to eliminate the variable ${}x_{1}$, then we look at an equation, say ${}G_{1}$, where ${}x_{1}$ occurs with a coefficient different from ${}0$. In this equation, we isolate ${}x_{1}$ and get a new equation of the form

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

where ${}x_{1}$ does not occur in ${}F_{1}$. Then in all other equations ${}G_{2},\ldots ,G_{m}$, we replace the variable ${}x_{1}$ by ${}F_{1}$, and obtain (after some simplifications) a linear system ${}G_{2}',\ldots ,G_{m}'$ without the variable ${}x_{1}$, which is, together with ${}G_{1}'$, equivalent with the original system.