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:

$$$\left\{ \begin{array}{c} x+y+z=3 \\ y-z=2 \\ z=-1 \end{array} \right.$$$

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,

$$$\left\{ \begin{array}{c} x+y+z=4 \\ y+z=2 \end{array} \right.$$$

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

$$z=\lambda \\ y=2-\lambda \\ x=2$$