Understanding Solutions to Systems of Linear Equations

[roam]

Homework Statement

This is from a solved problem:

Here is a system:

\begin{bmatrix} {1 & 0&0 \\ -1&0&0\\3&-5&0 \end{bmatrix} \begin{bmatrix} {x \\ y\\z \end{bmatrix}= \begin{bmatrix} {0\\ 0\\0\end{bmatrix}

A general solution of this system is

x= \begin{bmatrix} {0 \\ 0\\t \end{bmatrix}= \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} t

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

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.

 

[Mark44]

Homework Statement

This is from a solved problem:



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

The Attempt at a Solution

The matrix corresponds to the set of equations:

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

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.

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.