Linear Algebra - Solution Sets of Linear Systems

[GreenPrint]

See Attachment 2 for question or read below
Describe and compare the solution sets of X_1 + 5 X_2 - 3 X_3 = 0 and X_1 + 5 X_2 - 3 X_3 = -2.

See Attachment 1 for answer from back of book

I do not understand how the answer in the back of the book answers the question or were to even begin to get that answer. I do not see how I am suppose to come up with the solution to an equation with three variables and one equation. Am I suppose to assume that the X_1 X_2 X_3 are the same in both equations?
[1 5 -3 0
1 5 -3 -2]
~
[1 5 -3 0
0 0 0 -2]
system is inconsistent

Not exactly sure what I'm suppose to do to answer this question or what exactly I'm being asked. Thanks for any help anyone can provide.

 

[Dick]

Yes, you have to assume X1, X2 and X3 represent the same variables in the two equations. It might have been better if they had labeled them (x,y,z) instead of (X1,X2,X3). x+5y-3z=0 is a plane through the origin in (x,y,z) coordinates. x+5y-3z=(-2) is a plane that doesn't pass through the origin but is parallel to the first plane. It's displaced by a constant vector from the first plane, so they don't intersect, as you found. Take another look at the answer now and see if it makes more sense.

 

[GreenPrint]

Ok well the concept is making perfect sense to me know but I'm unsure on how to get the vectors u,v,p as they did. The vector p is <-2,0,0>, is it just by coincidence that the vector that the plates are displaced by is the last entry in the second row of as I could reduce the original matrix?

I'm understand the concept now a bit. The planes are defined by the base vectors u and v. I have to find u and v and the vector connects the two separate parallel planes which are defined by the equations. I'm still unsure how to proceed

 

[GreenPrint]

Don't I need more equations to define this form

Ax = 0
A = [1 5 -3]
x = [x y z]

Ax = b
A(p + tv) = b
A = [1 5 -3]
b = -2

hmm idk

 

[GreenPrint]

oh i got that was a mean problem lol

 

[Dick]

oh i got that was a mean problem lol

The vectors u, v and p that they give are not unique. u and v could be ANY basis for the plane that goes through the origin. And p could be ANY vector that is a difference between points in the two planes. That's maybe what's confusing about the problem.