Write the complete solution as the particular solution plus any multip

1. Feb 24, 2013

s3a

1. The problem statement, all variables and given/known data
Could someone please help me do this problem?:

“Write the complete solution as x_p plus any multiple of s in the nullspace:
x + 3y + 3z = 1
2x + 6y + 9z = 5
–x – 3y + 3z = 5”

The answer is x = x_p + x_n = {{-2},{0},{1}} + x_2 {{-3, 1, 0}}.

2. Relevant equations
Ax = b

3. The attempt at a solution
I get the {{-2},{0},{1}} and {{-3},{1},{0}} but, I don't get how the x_2 was obtained there.

Any input would be really appreciated!

2. Feb 24, 2013

LCKurtz

Re: Write the complete solution as the particular solution plus any mu

Did you try solving the system by row reduction?

3. Feb 24, 2013

tiny-tim

hi s3a!

(try using the X2 button just above the Reply box :
i've no idea why they've written x2

we'd normally use C (for "constant"! )

(btw, why are you using so many curly brackets, instead of fewer, ordinary brackets?)

4. Feb 24, 2013

LCKurtz

Re: Write the complete solution as the particular solution plus any mu

Also, are you sure the problem is copied correctly? I don't think your solution works.

 Yes it does work, arithmetic mistake here.

Last edited: Feb 24, 2013
5. Feb 24, 2013

s3a

Re: Write the complete solution as the particular solution plus any mu

Sorry. I double-posted. This website should have some sort of mechanism where you can't post twice until 10 seconds elapse or something like that.

6. Feb 24, 2013

s3a

Re: Write the complete solution as the particular solution plus any mu

LCKurtz, I did use row reduction. Also, thanks for telling me about the existence of the x_2 button and, yes, the solution was copied down correctly. Letting c = 1, for example, yields a solution to the system of equations so, it works. :)

tiny-tim, the abundance of curly brackets is Wolfram Alpha's notation for emphasizing that the vectors are vertical.

Okay, so basically x_2 = c is a constant. That pretty much helped me see through my confusion, lol!

Thanks!

7. Feb 24, 2013

LCKurtz

Re: Write the complete solution as the particular solution plus any mu

So you solved it OK with row reduction and all is well, right?

8. Feb 24, 2013

s3a

Re: Write the complete solution as the particular solution plus any mu

Yes.

To elaborate, I solved it augmented with b and that allowed me to get a particular solution and then took that fully-row-reduced matrix and took away the last column (=the augmented column) and replaced it with zeroes and that gave me the other part which, when all multiples of it are summed with the particular solution, all solutions are obtained.

(Sorry for the sloppy explanation, I am sleepy as I write this.)

(If anyone knows what that other part is called, I'd be interested in knowing.)

9. Feb 24, 2013

Dick

Re: Write the complete solution as the particular solution plus any mu

The part you got with the specific vector is called a 'particular solution'. The part with the parameter you got from Ax=0 is the 'homogeneous solution'.

10. Feb 24, 2013

s3a

Re: Write the complete solution as the particular solution plus any mu

Oh, ya ... I remember that term! Thanks!

11. Feb 24, 2013

LCKurtz

Re: Write the complete solution as the particular solution plus any mu

Of course, you don't have to solve it twice like that. For example if your reduced augmented matrix came out$$\begin{bmatrix} 1&3&0&-2\\ 0&0&1&1\\ 0&0&0&0 \end{bmatrix}$$that corresponds to the two equations
x + 3y = -2
z = 1
So you can let y be anything, say y = c and write the solution like this$$\begin{bmatrix} x\\ y\\ z\end{bmatrix}= \begin{bmatrix} -2-3c\\ c\\ 1 \end{bmatrix}=\begin{bmatrix} -2\\ 0\\ 1 \end{bmatrix}+c\begin{bmatrix} -3\\ 1\\ 0 \end{bmatrix}$$and you have it.

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