This is the situation of no solution. The last row gives us the equation

which is clearly an impossible situation.

If you attempted to solve the system of equations that led to the above augmented matrix using substitution, you might obtain along the way something like:

x1 + 2x2 + 5x3	=    2
−2x2 + 3x3	=    7
4x2 − 6x3	=  −7

If you solve the second of these for x₂ and substitute it into the third, the x₃'s subtract out, and you get:

and

and

therefore

which is an indication of an impossibility and hence no solution!

On the other hand, something like:

is not impossible. The last row produces an equation:

This situation indicates an infinite number of solutions.

We can set the last variable, say x₃, to any Real Number, t:

Then we can solve for the other two variables by substituting this into the equation for the second row to get x₂ and finally substituting into the equation for the first row to obtain x₁:

x2  =  −7/2 + 3/2•t

x1  =  9 − 8t

Another way of indicating this would be:

which is read: "the set of all triplets, (x₁,x₂,x₃), such that ....; where t is any Real value"