What does a linear equation with more than 2 variables represent? A linear equation with 2 variables, say $Ax+By+C = 0$, represents a line on a plane but what does a linear equation with 3 variables $Ax+By+Dz+c=0$ represent? A line in space, or something else? 
On a general note, what does a linear equation with $n$ variables represent?

In Geometry: the linear equation: $Ax+By+Dz+c=0$ in three variables $x$, $y$ & $z$ generally represents a plane in 3-D co-ordinate system having three orthogonal axes X, Y & Z. 
The constants $A$, $B$, $D$ shows the $\color{#0ae}{\text{direction ratios}}$ of the vector normal to the plane: $Ax+By+Dz+c=0$. Its direction cosines are given as $$\cos\alpha=\frac{A}{\sqrt{A^2+B^2+D^2}}, \quad\cos\beta=\frac{B} {\sqrt{A^2+B^2+D^2}} \quad \text{&} \quad\cos\gamma=\frac{D}{\sqrt{A^2+B^2+D^2}}$$ 
It can also be written in the $\color{#0ae} {\text{intercept form}}$ as follows $$\frac{x}{\left(\frac{-c}{A}\right)}+\frac{y}{\left(\frac{-c}{B}\right)}+\frac{z}{\left(\frac{-c}{D}\right)}=1$$ Where, $\left(\frac{-c}{A}\right)$, $\left(\frac{-c}{B}\right)$ & $\left(\frac{-c}{D}\right)$ are the intercepts of the plane with three orthogonal axes x, y & z respectively in the space.  
In Linear Algebra: the linear equation: $Ax+By+Dz+c=0$ represents one of the three linear equations of a system having unique solution, infinite solutions or no solution.   
