Linear algebra: nonhomogeneous system

1. The problem statement, all variables and given/known data
Let A = [tex]\left( \begin{array}{l} \begin{array}{*{20}c} 0 & 1 & { - 1} \\ \end{array} \\ \begin{array}{*{20}c} 2 & 1 & 1 \\ \end{array} \\
\end{array} \right)[/tex]. Suppose that for some b in [tex]\mathbb{R}^2[/tex], [tex]p = \left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\ \end{array} } \right)[/tex] is one particular solution to the nonhomogeneous equation Ax = b. What is the general form of the solution to this equation (Ax = b)?


2. Relevant equations
A relevant definition: A nonhomogeneous system is one with Ax = b, b is not equal to zero. A is a matrix and x and b are vectors.


3. The attempt at a solution
Since a homogeneous equation is Ax = 0, b = 0 in this particular question. Given p, I can presume that the general form of the solution will include p, but I'm not sure how to proceed.

 

The solution is:

[tex]\left( {\begin{array}{*{20}c}
1 \\
{ - 1} \\
1 \\

\end{array} } \right) + x_3 \left( {\begin{array}{*{20}c}
{ - 1} \\
1 \\
1 \\

\end{array} } \right),{\text{ }}x & _3 \in \mathbb{R}[/tex]

But I can't make sense out of it.

 

I'm guessing since it's in the solution, the solution space is contained in R, but I'm not sure how to verify that it is actually a family of solutions. I'm trying to understand the presentation of the solution in terms of two vectors, one being multiplied by x3 for some reason.