TITLE: Geometric Interpretation of Solutions to Linear Systems
QUESTION [5 upvotes]: I've been reading Linear Algebra by Jacob for self study and I'm wondering about the geometric interpretation of a unique solution to systems of equations in $\mathbb R^3$. 
For example, the homogeneous system $$ 
\begin{eqnarray*}
x+5y-2z &=&0, \\
x-3y+z &=& 0, \\ 
x+5y-z &=& 0 
\end{eqnarray*}
$$ has the unique solution $x=y=z=0$. From Calc III, I would think of this as being 3 planes and the various $(x,y,z)$ values of the line (vector) that forms the intersection as the solution to the system. But the solution here and in many other systems in $\mathbb R^3$ is a point and that doesn't seem to be possible. 
Am I wrong in trying to understand this geometrically or is there something I'm missing? 
Sorry if this is hard to read, I haven't learning latex yet and thanks to everyone who responded to my question about self study books for real analysis.

REPLY [2 votes]: When you intersect three planes in $\mathbb{R}^3$, you can get a plane, a line, a point, or the empty set. 

You get the empty set if two of the planes are parallel but distinct.
You get a plane if the three planes are identical. 

In all other cases two of the planes are non-parallel, so they intersect in a straight line; call it $L$. 

If that line is in the third plane, the intersection of all three planes is the line $L$. 
If $L$ is parallel to the third plane, the intersection of the three planes is empty. 
And if $L$ cuts through the third plane, the intersection of the three planes is a point, the point of intersection of $L$ and the third plane.