How to express solution to system of equations

[dewert]

Sorry if this is too basic for this forum, but it IS from a 2nd-year linear algebra course. I'm probably just being stupid, and missing something obvious, but here goes:

Homework Statement

Solve the following system of equations:

$$2x_{1} - 2x_{2} - 3x_{3} = -2$$
$$3x_{1} - 3x_{2} - 2x_{3} + 5x_{4} = 7$$
$$x_{1} - x_{2} - 2x_{3} - x_{4} = -3$$

The attempt at a solution

So, clearly there won't be a unique solution. I do the work and get this matrix:
$$\[ \left( \begin{array}{cccc|c} 1 & -1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right)\]$$

However, the answer in the back of the book is
{r(1,1,0,0) + s(-3,0,-2,1) + (5,0,4,0) | r,s $$\in \mathbb{R}$$}

My question is how to get this from the solution matrix. Regardless of whether the solution is right or wrong, I don't know where this is coming from. I can conceivably see that (5,0,4,0) can be obtained by choosing x_2 = 0, but that's about it.

Thanks!

 

[CompuChip]

Are you sure you don't get something else, like, say,
$$\[ \left( \begin{array}{cccc|c} 2 & -2 & 0 & 3 & 5 \\ 0 & 0 & 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array} \right)\]$$