How to arrive at correct parametric form In question 2 of this quiz, the resulting parametric vector form is $x = x_3 \begin{align}
    \begin{bmatrix}
           1 \\
           1 \\
           1 \\
         \end{bmatrix}
  \end{align}$. How do they arrive at $x = x_3 \begin{align}
    \begin{bmatrix}
           1 \\
           1 \\
           1 \\
         \end{bmatrix}
  \end{align}$?
I reduced the augmented matrix given is to $
\left[
    \begin{matrix}
    1 & 0 & -1 & 0\\
    0 & 1 & -1 & 0\\
    0 & 0 & 0 & 0\\
    \end{matrix}
\right]
$ from that I get $x_1 = x_3$ and $x_2 = x_3$.
I then get the parametric vector form of $x_1\begin{align}
    \begin{bmatrix}
           1 \\
           0 \\
           0 \\
         \end{bmatrix}
  \end{align}$ + $x_2\begin{align}
    \begin{bmatrix}
           0 \\
           1 \\
           0 \\
         \end{bmatrix}
  \end{align}$ + $x_3\begin{align}
    \begin{bmatrix}
           -1 \\
           -1 \\
           0 \\
         \end{bmatrix}
  \end{align}$ = $\begin{align}
    \begin{bmatrix}
           0 \\
           0 \\
           0 \\
         \end{bmatrix}
  \end{align}$
From that please help me understand how to arrive at $x = x_3 \begin{align}
    \begin{bmatrix}
           1 \\
           1 \\
           1 \\
         \end{bmatrix}
  \end{align}$. Thank you.

If $x=(x_1,x_2,x_3)^T$ is a solution, then you already commented that you know that $x_1=x_2=x_3$. But then
$$
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}= \begin{pmatrix} x_3 \\ x_3 \\ x_3\end{pmatrix}= x_3 \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}
$$
