Matrices - solution sets: find all solutions to $Ax = 0$

Question

I'm asked to find the solutions to the system below. I have the full solution here, but I don't quite understand it. I was wondering if someone could explain where the values in the two tuples multipled by s and t came from? I circled it in the picture below.

picture of question

Answer 1

I take it you know how you find the general solution of a matrix ewuation Ax=0, by first reducing to row echelon form.

The matrix you have has 2 leading ones (look at this in case you do not know what this term means: https://en.m.wikipedia.org/wiki/Row_echelon_form).

Since $x_1$ is a leading one, you rearrange to express it in terms of the other variables, ie you get that $x_1=- 2x_2-x_4$.

Solving for $x_3$,ie the second leading one you get that $x_3=0$.

So the general solution is:

$x_1=-2x_2-x_4$

$x_2=x_2$

$x_3=0$

$x_4=x_4$

$x_5=x_5$.

Which gives the answer mentioned in the book.