# Homework Help: General solution of linear system

1. Jul 25, 2010

### estro

I have this question, but don't know how to even start.

Suppose (M) is a linear system of 2 equations and 3 unknowns, where (2,-3,1) its solution.
Suppose (O) is a matching homogeneous linear system, where (-1,1,1) and (1,0,1) its solutions.

How can I find the general solution of (M)?
I'm totally lost with this one and appreciate any help.

Last edited: Jul 25, 2010
2. Jul 25, 2010

### Staff: Mentor

Where (2, -3, 1) is what?
Is there a word missing here?
I understand what you're trying to say, but I would like you to rephrase things so that you have complete thoughts.

3. Jul 25, 2010

### estro

Edited the first post. Sorry for the missing words (I guess 24+ hours without sleep make me skip words).

Last edited: Jul 26, 2010
4. Jul 26, 2010

### HallsofIvy

Still missing words. Also, you cannot say that these are the solutions to the homogeneous system nor that (2, -3, 1) is the solution to the original system. Such a problem has an infinite number of solutions. If you can find a complete set of "independent" solutions then you can write any solution to the homogenous system as a linear combination of them. What is given here is not sufficient to conclude that there isn't a third linearly independent but to get an answer to this, we must assume so. It would have been a lot better if you had simply copied the problem as it was given.

Anyway, assuming that any solution to the homogeneous system must be of the form A(-1, 1, 1)+ B(1, 0, 1) for some numbers A and B. Notice that L(A(-1, 1, 1)+ B(1, 0, 1))= AL(-1, 1, 1)+ BL((1, 0, 1))= A(0)+ B(0)= 0. Further, if x is a solution to the original system, if L(x)= y where y was the given "right side" of the system, then L(A(-1, 1, 1)+ B(1, 0, 1)+ x)= AL(-1, 1, 1)+ BL(1, 0, 1)+ L(x)= 0+ 0+ L(x)= y.

5. Jul 26, 2010

### estro

Sorry for my grammar, English is not my native tongue.
Anyway thanks for the hint.