# How to know if the solution for an underdetermined linear system fills entire space?

1. Sep 24, 2008

### Brutus

1. The problem statement, all variables and given/known data
Solve:
$$\left\{ \begin{array} {ccc} 5x - y - z & = & 4 \\ x - y + 2z & = & -5 \end{array} \right.$$

If the system has infinite solution, does it fill the entire space?

3. The attempt at a solution

I know the system is underdetermined since the number of variables is less than the number of given equations, so my first step was to find out whether this system is consistent(has at least one solution).

$$5x - y - 4 & = & \frac{-x}{2} + \frac{y}{2} - \frac{5}{2} \\ y & = & \frac{11x}{3} - 1$$

The system is consistent and has infinite solution.

Now, the real question, is the solution constrained to a plane or does it fill the entire space?
How do you find out? I've thought about it, and I am completely lost.

Last edited: Sep 24, 2008
2. Sep 24, 2008

### Defennder

Re: How to know if the solution for an underdetermined linear system fills entire spa

The first thing you should do is to actually solve for the expression giving you the general form of the solution to the equations. You can tell at a glance that it will have an infinite number of solutions, since it isn't inconsistent. Write it out using a parameter t. Then ask yourself if you recognise the geometric significance of the general solution. That answers your question.

3. Sep 24, 2008

### Brutus

Re: How to know if the solution for an underdetermined linear system fills entire spa

Ok, I got it now.

It is a line, the intersection of two planes, haha, boy I feel stupid now...

Thanks.