How are the x1 and x4 values determined in the solution to the matrix equation?

[kostoglotov]

Hi, rapid fire posting in this subforum I know, sorry if that's annoying. Let me know if I should space my posts out a bit more.

Here's an image of the solution to a worked example (from Intro to Linear Algebra 4th by Strang)

here's the imgur link: http://i.imgur.com/IG6r15H.jpg

I cannot figure out what the underlined (underlining added by me) sentences mean:

"The first and last rows give x₂ and x₃. Then the middle rows give x₁ and x₄."

How do the middle rows only give x₁ and x₄? Is this the result of treating this as a system of simultaneous equations rather than linear combinations? That's the only explanation I can find at the moment. Mostly I don't know what the author's intended meaning is.

I could find all the x's by solving simultaneous equations, but would that be how the author has done it, or has Strang found the x's a different way?

 

[BvU]

What the author describes IS the solving of the 'simultaneous' equations :

$x_2 = b_1 \\ x_3-x_1 = b_2\\ x_4 - x_2 = b_3\\ -x_3 = b_4$​

we humans don't do 'simultaneously: we see that this means that

first and last rows give x₂ and x₃ and THEN the middle rows give x₁ and x₄.​

Thereby Cx = b has been solved, and that can only be the case if C is invertible (x = C^-1 b).

 

[Orodruin]

Solving the system of simultaneous equations just means that the same solution should be valid for all equations.