Matrices, I can't seem to cancel out any more elements

[mr_coffee]

Hello everyone, every row operation I make now is taking out the 0's i just made, any ideas on what I can do from here?
Here is my work:
http://img201.imageshack.us/img201/5573/lastscan2ah.jpg

Also I was confused on how this works, my professor did an example:
3x+2y+z = 8;
x = a
y = b
z = 8 - 3a - 2b
I get this part but then he does the following:

http://img140.imageshack.us/img140/1337/eeeeeee1ut.jpg

Where is he getting those values? The first one looks like he let a = 0 and b = 0, then ur left with just 8, but I'm not sure.
Thanks.

 

[LeBrad]

You have 3 equations and 5 unknowns, so you will have 2 free variables. You can't reduce it any more than that.

 

[mr_coffee]

hm...What do i do from there then? How can you solve the system?

 

[LeBrad]

He's just writing it as a sum of vectors multiplied by some coefficient. For example,

3+x 3 1 0
x^2 = 0 *1 + 0 *x + 1 *x^2
X^2-4x+7 7 -4 1

or if you write it out as 3 separate equations, which it actually is,

3 + x = 3*1 + 1*x + 0*x^2
x^2 = 0*1 + 0*x + 1*x^2
x^2-4x+7 = 7*1 + -4*x + 1*x^2

 

[mr_coffee]

Why does it look like all he is doing is letting a and b equal 0 in the first one, which will make 0 0 8 then he's letting a = 1, and b = 0, and finding the values then, letting a = 0 and b = 1, and then finding the values?

 

[HallsofIvy]

Because that is what he is doing! There are 3 equations in 5 unknowns. You can solve for 3 of the unknowns (x,y,z) in terms of the other 2 but those 2 (a and b) can be anything. To write a general formula your teacher is saying, "First,suppose a= 1, b= 0. The x,y,z= something written as a vector v₁. Now, suppose a= 0, b=1. Okay, then x,y,z= something else written as a vector v₂. " The general formula is then av₁+ bv₂. That's a linear combination that obviously gives the correct answer when a=1, b=0 and when a=0, b=1 and that's enough to show it is the general solution to this linear problem.

(Oh, the bit about letting a= 0, b= 0 is because this is not a "homogeneous" problem.)

 

[mr_coffee]

Well if that is infact what's going on, then how is this possible?>
3x+2y+z = 8;
x = a
y = b
z = 8 - 3a - 2b

Okay he let a = 0, b = 0, so z = 8
a = 1, b = 0, he has z = -3
z = 8 -3(1) - 2(0) = 5?
then he has
a = 0; b = 1; z = -2
z = 8 -3(0) -2(1) = 6?
but he has -2.