Systems in echelon form

The systems in echelon form are those in that every equation has one unknown less than the previous one.

See the following example:

Example

${\begin{matrix} x + y + z = 3 \\ y - z = 2 \\ z = - 1 \end{matrix}$

It is simple to solve.

We start with $z = - 1$ and we replace it in the second equation. We obtain $y + 1 = 2$ , so $y = 1$ .

We substitute now in the first equation: $x + 1 - 1 = 3$ ; so $x = 3$ .

The solution is then $(3, 1, - 1)$ and it is unique.

Obviously it can happen that there are more unknowns than equations. The system will not have a unique solution. Lets have, for example,

Example

${\begin{matrix} x + y + z = 4 \\ y + z = 2 \end{matrix}$

In this case we will give to $z$ any value (which we will call $λ$ ) and follow the same procedure, substituting in the other equations. Therefore,

$z = λ y = 2 - λ x = 2$