Homework Help: System of Linear Equations

1. Jun 13, 2010

roam

1. The problem statement, all variables and given/known data

This is from a solved problem:

So, how did they get this solution? I tried solving this and I just don't get it!

3. The attempt at a solution

The matrix corresponds to the set of equations:

x=0
-x=0
3x-5y=0

If I take "y" to be the free variable, I have x=5/3y therefore the solution to the system will be

$$\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {5/3 \\ 1\\0 \end{bmatrix} y + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z$$

But if I take "x" to be the free variable I get y=3/5x, so I get the following solution:

$$\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {1 \\ 3/5\\0 \end{bmatrix} x + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z$$

How do I know which one (x or y) should be taken as the free variable?

Could anyone please explain to me how to get the right answer? I just can't see how to deal with this type of problems.

2. Jun 13, 2010

Staff: Mentor

From these equations, x is obviously 0. With x = 0, the 3rd equation implies that y = 0. This means that z is the free variable.