I simply cannot do this Gauss elimination question

**thomasashcroft** · August 15th 2009, 06:19 PM

$ 2x - y + 3z + t = 1 $
$ -2x + y - 2z = 1 $
$ 4x - 2y + 7z + 3t = 4 $

I got it to

(2 -1 3 1 1)
(0 0 1 1 2)

and I don't know what to do from there : /

**mr fantastic** · August 15th 2009, 06:51 PM

Originally Posted by thomasashcroft

$ 2x - y + 3z + t = 1 $
$ -2x + y - 2z = 1 $
$ 4x - 2y + 7z + 3t = 4 $

I got it to

(2 -1 3 1 1)
(0 0 1 1 2)

and I don't know what to do from there : /

Get the reduced row echelon form. Then it should be clear that you have to introduce two parameters .... Perhaps $t = \lambda$ and $x = \mu$ where each parameter can be any real number. Now get y and z in terms of these parameters and you have a solution.

**thomasashcroft** · August 15th 2009, 06:52 PM

Originally Posted by mr fantastic

Get the reduced row echelon form. Then it should be clear that you have to introduce two parameters .... Perhaps $t = \lambda$ and $x = \mu$ where each parameter can be any real number. Now get y and z in terms of these parameters and you have a solution.

It's not possible to get the reduced row echelon form is it?

**mr fantastic** · August 15th 2009, 07:02 PM

Originally Posted by thomasashcroft

It's not possible to get the reduced row echelon form is it?

Why do you think that? Of course it is.

PlanetMath: reduced row echelon form

**thomasashcroft** · August 15th 2009, 07:06 PM

Well all I would have to do is multiply the first row by 1/2 which would leave me with 1 - 0.5 1.5 0.5 0.5 in the first row.. and then it'd be in r.r.e.f. But how does that help me see that y should be arbitrary?

**mr fantastic** · August 15th 2009, 10:30 PM

Originally Posted by thomasashcroft

Well all I would have to do is multiply the first row by 1/2 which would leave me with 1 - 0.5 1.5 0.5 0.5 in the first row.. and then it'd be in r.r.e.f. But how does that help me see that y should be arbitrary?

$R_1 \longrightarrow R_1 - 3 R_2$ :

$\left( \begin{array}{cccc} 2 & -1 & 0 & -2 \\ 0 & 0 & 1 & 1 \end{array} \right|$ $\left. \begin{array}{c} -5 \\ 2 \end{array} \right)$

which is equivalent to the linear system:

$2x - y - 2t = -5$

$z + t = 2$

Now there is choice. I chose $t = \lambda$ and $x = \mu$ where both parameters can be any real number.

Then:

$z + \lambda = 2 \Rightarrow z = 2 - \lambda$ .

$2 \mu - y - 2 \lambda = -5 \Rightarrow y = 2 \mu + 3$ .

Other choices leading to equivalent solutions can obviously be made.

**CaptainBlack** · August 16th 2009, 10:43 AM

Originally Posted by thomasashcroft

$ 2x - y + 3z + t = 1 $
$ -2x + y - 2z = 1 $
$ 4x - 2y + 7z + 3t = 4 $

I got it to

(2 -1 3 1 1)
(0 0 1 1 2)

and I don't know what to do from there : /

The initial tablau is:

$ \begin{array}{rrrr|r} 2&-1&3&1&1\\ -2&1&-2&0&1\\ 4&-2&7&3&4 \end{array} $

add the first row to the second:

$ \begin{array}{rrrr|r} 2&-1&3&1&1\\ 0&0&1&1&2\\ 4&-2&7&3&4 \end{array} $

Now subtract twice the first row from the third:

$ \begin{array}{rrrr|r} 2&-1&3&1&1\\ 0&0&1&1&2\\ 0&0&1&1&2 \end{array} $

Removing the duplicate leaves you with:

$ \begin{array}{rrrr|r} 2&-1&3&1&1\\ 0&0&1&1&2 \end{array} $

NOw let $t=\alpha$ and $y=\beta$ and solve for $x$ and $z$ in terms of $\alpha$ and $\beta$ .

CB

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