Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Describe all solutions of the system (the last column is the augmented column)

$-x_1 +2x_2 + x_3 + 4x_4 = 0$

$2x_1 + x_2 -x_3 +x_4 = 1$

$\pmatrix{-1&2&1&4&0 \\ 2&1&-1&1&1 } \sim \pmatrix{1&-2&-1&-4&0 \\ 0&5&1&9&1} \sim \pmatrix{1&0 & -\dfrac{3}{5}& -\dfrac{2}{5}& \dfrac{2}{5} \\ 0&5&1&9&1} $

Solving we get the equations:

$x_1 = \dfrac{3}{5}x_3 + \dfrac{2}{5}x_4 + \dfrac{2}{5} \iff x_1 = 3x_3 + 2x_4 + 2$ (Can I scale like this?)

$5x_2 = -x_3 -9x_4 +1 \iff x_2 = x_3 +9x_4 - 1$ (can I scale like this?)

Now we have:

$\langle x_1, x_2, x_3, x_4\rangle = \langle 3x_3 + 2x_4 +2, -x_3 -9x_4 + 1, x_3, x_4\rangle$

$= x_3\langle 3, -1, 1, 0 \rangle + x_4\langle 2, -9, 0, 1\rangle + \langle 2,1,0,0\rangle $

Where did I go wrong?

share|improve this question
    
Where does $\pmatrix{-1&2&3&4&0 \\ 2&1&-3&4&1 }$ come from? Do you mean $\pmatrix{-1&2&1&4 \\ 2&1&-1&1 }\cdot \vec x= \pmatrix{0\\ 1}$? –  draks ... Oct 4 '12 at 21:11
    
Yes, I really wanted an augmented matrix –  CodeKingPlusPlus Oct 4 '12 at 21:14
    
Why do you add coloumns instead of rows? –  draks ... Oct 4 '12 at 21:26
    
I don't understand your scaling. The right hand equations aren't the same as the left, so they don't necessarily have the same solution space. –  Cocopuffs Oct 4 '12 at 21:53
add comment

2 Answers

up vote 1 down vote accepted

Actually it should start with:

$\pmatrix{-1&2&1&4&0 \\ 2&1&-1&1&1}$

In response to your (can I scale like this) question, no. It is an equation and to preserve the equality, all elements should be scaled, not just one side of the equation.

share|improve this answer
    
We both copied the system into the matrix incorrectly... Or you were just copying me... –  CodeKingPlusPlus Oct 4 '12 at 21:40
    
I copied yours, but see above for what it should be. –  adam W Oct 4 '12 at 21:42
    
Yes, I copied it down wrong from paper. However, I copied down the equations correctly. Why do they not work? –  CodeKingPlusPlus Oct 4 '12 at 21:48
    
It is not solved correctly yet, see above; do not scale only part of an equation. –  adam W Oct 4 '12 at 21:52
    
Thanks, that was the issue! Could you also explain how I can check this solution? –  CodeKingPlusPlus Oct 4 '12 at 21:54
show 1 more comment

You need to be careful scaling.

You have the equations: $$-x_1+\frac{3}{5}x_3 + \frac{2}{5}x_4 = -\frac{2}{5}, \ \ \ 5 x_2 + x_3 + 9x_4 = 1$$ which can be written as $$x_1 = \frac{3}{5}x_3 + \frac{2}{5}x_4 + \frac{2}{5}, \ \ \ x_2 = -\frac{1}{5} x_3 - \frac{9}{5} x_4 + \frac{1}{5}$$ Thus all the solutions are: $$x = \frac{1}{5} \pmatrix{3 \\ -1 \\ 5 \\ 0} x_3 + \frac{1}{5} \pmatrix{2 \\ -9 \\ 0 \\ 5} x_4 + \frac{1}{5} \pmatrix{2 \\ 1 \\ 0 \\ 0}$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.