Linear algebra: System of equations problem

Question

I have been doing one exercise and got a problem. The exercise is as follows:

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Now, in the beginning of solutions it says that one should first find the solution space V for the system of equations: photo2 And that the solution space V is consisted of all vectors v of form:

I do not understand how did they get this form for the vector v. If I do Gauss Jordan on the equation system, I get a different solution. This is one point of this task that I do not understand. The next steps for getting to the final solution are easy. (Please help me get a few more rep so that I can post photos, not links, thanks)

Answer 1

Guide:

$$x_1-x_2-x_3-3x_4=0$$

$$x_1-2x_2+x_3+x_5=0$$

Let $x_2=r, x_3=s, x_4=t$, solve for $x_1$ and $x_5$ in terms of $r,s,t$.