Relation between sets of solutions Consider a non-homogeneous linear system $Ax=b$ with four equations and three variables. Call $(Σ)$ the corresponding homogeneous system. What can we conclude about the general solutions of the two systems, given that $rank(A)=rank([A\mid  b])=2  $ ?
Here's what I've thought so far:
Since $A$ and $[A \mid b]$ have the same rank we know that $Ax=b$ has a solution. 
It can also be proven that if $z$ and $y$ are two solutions of $Ax=b$ these are related by the equation $y=c+x$ where $c$ is a solution of $(Σ)$. I don't know if any other conclusions can be made. Any help will be appreciated.
 A: We can additionally specify the set of all solutions. In fact, the essence can already be seen when considering the corresponding homogeneous linear system.

Let $A$ be the coefficient matrix of a homogeneous, linear system of equations with $n$ variables
\begin{align*}
Ax=0\tag{1}
\end{align*}
The following is valid:
  
  
*
  
*The set of all solutions is a subspace $U$ of $K^n$, with $\mathop{dim}U=n-\mathop{rank}(A)$.
  
*If $\mathop{rank}(A)\neq n$, there are $s=n-\mathop{rank}(A)$ linear independent solution vectors $c_1,\ldots,c_s$ and
  \begin{align*}
x=\lambda_1 c_1+\cdots +\lambda_s c_s\tag{2}
\end{align*}
  with arbitrary $\lambda_1,\ldots,\lambda_s\in K$ is the general solution.

Since in our example the rank of $A$ is two and there are three variables, the set of all solutions is a subspace $U$ of $K^3$ with dimension $$\mathop{dim}U=3-\mathop{rk}(A)=1.$$ $K$ is the underlying field, typically $\mathbb{R}$ or $\mathbb{C}$.
So, if we find a specific solution $x=c_1$ of (1), the set of all solutions is given by
\begin{align*}
x=\lambda_1 c_1\qquad\qquad \lambda_1\in K
\end{align*}

Now, let's consider the non-homogeneous linear system of equations $Ax=b$ with $b\neq 0$.
OP has already noted, when solutions do exist ($\mathop{rank}(A)=\mathop{rank}([A\mid b])$) and how different solutions of the inhomogeneous linear system are related. Based upon these facts and (2) we can describe the set of all solutions of the inhomogeneous linear system of equations.
Let $A$ be the coefficient matrix of an non-homogeneous, linear system of equations with $n$ variables
\begin{align*}
Ax=b
\end{align*}
The following is valid:
  
  
*
  
*If $s=n-\mathop{rank}(A)>0$ and $B=\{c_1,\ldots,c_s\}$ is a basis of the vector space of solutions of the corresponding homogeneous linear system of equations (1), the general solution of the non-homogeneous linear system of equations has the form
  \begin{align*}
x=c_0+\lambda_1 c_1+\cdots +\lambda_s c_s
\end{align*}
  with $c_0$ a solution of $Ax=b$ and arbitrary $\lambda_1,\ldots,\lambda_s\in K$.
  

In our example we see, that the set of all solutions is given by
\begin{align*}
x=c_0+\lambda_1 c_1\qquad\qquad \lambda_1\in K
\end{align*}
with $c_0$ a specific solution of $Ax=b$.
