Gaussian Elimination: Only one nonzero row

Question

I was given the following system of equations:

(x1) + 2(x2) - (x3) + (x4) = 1

2(x1) + 4(x2) - 2(x3) + 2(x4) = 2

5(x1) + 10(x2) - 5(x3) + 5(x4) = 5

When obtaining the row-echelon matrix of this system, I get:

1 2 -1 1 | 1

0 0 0 0 | 0

0 0 0 0 | 0

I am left with (x1) + 2(x2) - (x3) + (x4) = 1

Is there no way I can conclude the solution set of this system of equations?

Thank you for the help!

Answer 1

Let $x_2=r, x_3=s, x_4=t$, then $x_1=1-2r+s-t$.

The solution is characterized by 3 parameters, $r,s,$ and $t$,

$$\left\{(1-2r+s-t,r,s,t) |r,s,t \in \mathbb{R}\right\}$$