A question about "General and Particular Solutions" (I guess) which I'm not sure what they mean We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: 
What is the number of free variables in the solution to the system $Ax = 0$? 
(Well, it must be 3. Although I think I know the answer, I didn't exclude this part of question because I think there must be somebody who can explain better.) 
For a given $b$, are we guaranteed to have a solution to $Ax = b$? 
If we have a solution, what is the dimension of the solution space? 
(The asker wants us to explain without using an example.)
 A: You seem to think that since the question requires a solution without examples, you may not use examples to solve the problem.  Not true! Playing with an example can give you the insight to understand the concept behind the problem.
The simplest $3 \times 6$ matrix that I can think of with rank 3 is
$$
A = \begin{bmatrix}
    1 & 0 & 0 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 & 0 \\
    \end{bmatrix}
$$
Then any vector of the form $x_0 = \begin{bmatrix} 0 & 0 & 0 & x_4 & x_5 & x_6 \end{bmatrix}^T$ is a solution to $Ax=0$.  We see three free variables.  
Given a vector $b$ in $\mathbb{R}^3$, the system $Ax=b$ has a solution if $b$ can be written as a linear combination of columns of $A$.  If $b = \begin{bmatrix} c_1 & c_2 & c_3 \end{bmatrix}^T$, then $x_p = \begin{bmatrix} c_1 & c_2 & c_3 & 0 & 0 & 0 \end{bmatrix}^T$ will do (try it).  In fact, $\begin{bmatrix} c_1 & c_2 & c_3 & x_4 & x_5 & x_6 \end{bmatrix}^T$ is also a solution for any values of $x_4$, $x_5$, and $x_6$.  Therefore, solution space is an affine space of dimension 3.
In general, the rank-nullity theorem says the rank plus the nullity (dimension of the kernel) of a matrix equals the number of columns in the matrix.  If $A$ is of rank 3, its columns span $\mathbb{R}^3$.  Finally, if $x_0$ is a solution to $Ax=0$, and $x_p$ is a solution to $Ax=b$, what can you say about $x_0 + x_p$?
PS In your question title you ask about the terms general and particular solution.  In a system of equations of the form $Ax=b$, a particular solution is a single solution, and the general solution is a description of all solutions.  What this problem is illustrating is that once one particular solution is found, any solution can be found by adding to the particular solution a solution to the homogeneous system $Ax=0$.
