Free Variables Interpretation Suppose I have a system of equations $At=b$ representing two planes in the euclidian space $\mathbb{R}^3$ where $A$ is $2\times 3$ and $t$ is $(x y z)$.    
The matrix of the system has a 1-dimension NullSpace, that means the intersection of the planes shifted at the origin is a line.If the only possible free variables are x and y for example, does it mean the line is in the $x,y$ plane ?     
If the only possible free variables are $x$ and $z$ for example, does it mean the line is in the $x,z$ plane?
As an extension of my doubt, if for example we had two hyperplanes ( $x,y,z,w$ as variables ) that intercepted in a plane and the only possible pair of free variables was $(x,y)$ and $(x,z)$ would that mean that the plan would not span the $w$ dimension , only the $x,y,z$ dimension ?
 A: Yes, you are correct.  Here is an outline of the possibilities:
A Line in $\mathbb{R}^2$
For a line in $\mathbb{R}^2$, usually either $x$ or $y$ can serve as a free variable.  The exception is a horizontal line, for which only $x$ can be free, or vertical lines, for which only $y$ can be free.
A Line in $\mathbb{R}^3$
Something similar holds for a line in $\mathbb{R}^3$.  There are three cases:


*

*Usually any of $x$, $y$, or $z$ can be a free variable.

*However, if the direction of the line is perpendicular to one of the axes, then that variable can no longer be free.  For example, if a line is horizontal (perpendicular in direction to the $z$ axis, and parallel with the $xy$-plane), then $z$ is not an allowed free variable.

*The last case is that the line is parallel to one of the axes—say the $x$-axis—in which case $x$ is the only possible free varaible.
A Plane in $\mathbb{R}^3$
For a plane in $\mathbb{R}^3$, there are again three cases:


*

*Usually any two of the variables can be free.

*However, if the plane is perpendicular to one of the coordinate planes, then that pair of variables can't be free.  For example, if a plane is vertical (perpendicular to the $xy$-plane), then $\{x,y\}$ is not an allowable set of free variables.

*Finally, if a plane is parallel to one of the coordinate planes, then that is the only allowed set of free variables.  For example, if a plane is horizontal (parallel to the $xy$-plane), then $\{x,y\}$ is the only possible set of free variables.
Things in $\mathbb{R}^4$
Lines and hyperplanes in $\mathbb{R}^4$ have four possibilities.  For a line, either any variable is allowed, or three are allowed, or two are allowed, or only one is allowed.  For a hyperplane, either any three are allowed, or one triple is excluded, or a certain pair is excluded (e.g. $x$ and $y$ can't simultaneously be free), or there is only one allowed triple.
Things get complicated for two-dimensional planes in $\mathbb{R}^4$.  There are seven different possibilities.  Here is an example of each:


*

*Any two variables are allowed.  This is the most common situation.

*Any two of $\{x,y,z\}$ are allowed, but $w$ is not allowed.  This happens when the direction of the plane is perpendicular to the $w$-axis.

*Only $x$ and $y$ are allowed, and neither $z$ or $w$ can be free.  This happens when the plane is fully perpendicular (orthogonal) in direction to the $zw$-plane.

*Any two except $\{x,y\}$ are allowed.  This happens when the plane contains a line whose direction is perpendicular to the $xy$-plane.

*Either $x$ or $y$ or $z$ is allowed (but not two of these), and $w$ is required.  This happens when the plane is perpendicular to the $xyz$-subspace.

*Either $x$ or $y$ is allowed (but not both), $z$ is required, and $w$ is not allowed.  This happens when the direction of the plane is perpendicular to the $w$-axis and the plane contains a line that's perpendicular in direction to the $xy$-plane.

*Either $x$ or $y$ is allowed (but not both), and either $z$ or $w$ is allowed (but not both).  This happens when the plane contains a line perpendicular in direction to the $xy$-plane, and also contains a line perpendicular in direction to the $zw$-plane.
By the way, the possibilities listed here are known as linear matroids.  They correspond to the possible dependence relations between the columns of a matrix with a certain rank.  (In this case, the columns of a 4-column matrix with rank 2.)
