Two systems of linear equations are said to be **equivalent** when they have exactly the same solutions. For instance

$$ \left\{ \begin{array}{rrrrr} x_1 & -2x_2 & + x_3 & = & 7\\ 2x_1 & -5x_2 & +2x_3 & = & 6\\ 3x_1 & 2x_2 & - x_3 & = & 1\\ \end{array}\right. \qquad\qquad \left\{ \begin{array}{rrrrr} 2x_1 & -5x_2 & +2x_3 & = & 6\\ x_1 & -2x_2 & + x_3 & = & 7\\ 3x_1 & 2x_2 & - x_3 & = & 1\\ \end{array}\right. $$

are equivalent, because we have just swapped the two first rows, and the requirements to fulfill are the same. On the other hand,

$$ \left\{ \begin{array}{rrrrr} x_1 & -2x_2 & + x_3 & = & 7\\ 2x_1 & -5x_2 & +2x_3 & = & 6\\ 3x_1 & 2x_2 & - x_3 & = & 1\\ \end{array}\right. \qquad\qquad \left\{ \begin{array}{rrrrr} x & & & = & 2\\ & y & & = & 8\\ & & z & = & 21\\ \end{array}\right. $$

are equivalent too (and thus both systems have only one solution $(x=2,y=8,z=21)$), but this is not so obvious for the naked eye

Our strategy to solve systems of linear equations will be to get simpler and simpler equivalent systems, until we reach a system whose solutions are straightforward to find