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,

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