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

${G_1}^{\prime }:x_1=F_1,$

where $x_1$ does not occur in $F_1$. Then in all other equations $G_2,\dots ,G_m$, we replace the variable $x_1$ by $F_1$, and obtain (after some simplifications) a linear system ${G_2}^{\prime },\dots ,{G_m}^{\prime }$ without the variable $x_1$, which is, together with ${G_1}^{\prime }$, equivalent to the original system.