Hello, I have the following equations: X 1 - 2X 2 + 3X 3 = b 1 -X 1 + X 2 -2X 3

Question

Hello,
I have the following equations:

X1 - 2X2 + 3X3 = b1

-X1 + X2 -2X3 = b2

2X1 - X2 + 3X3 = b3

After reducing the augmented matrix, we have b1+3b2+b3=0 as the condition for the system to have solutions.

and they say (-4,-3,0) is the solution of the nonhomogeneous system
while alpha times (-1,1,1) is the most general solution of the homogenous system corresponding to the system of equations listed above ( and I don't get this part, how can they assume values for the b's and then claim that this is the general solution, and how do I know which is the solution for the homogenous part and which is the solution of the non homogenous)

Can you please explain that to me? Thanks. [details would be appreciated]

Explanation / Answer

homogeneous system of linear equations is when b1=b2=b3=0. put those values in the equations, do gaussian elimination and you get the solutions corresponding to the homogeneous system, v=(-1,1,1). Now assume any values for b1,b2,b3. ( only condition required is that the resultant system should have atleast one solution.). Find a particular solution to that set up,p=(-4,-3,0) here. Then the general solution to all non homogeneous systems of the form Ax=b with any values for b's can be found as s=p+av. where a is any constant. Basically the solution set for Ax=b is a linear translation of the solution set for Ax=0.

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