A line in three-dimensional space may be described as an intesection of two planes, for example: $\begin{align}x+y+z=0\tag{1}\\3x+7y=1\tag{2}\end{align}$ This can be understood as two separate scalar equations or as a single matrix equation. (One may also describe a line parametrically.)
This poses a question: is it possible to express the same line using a single scalar equation? It turns out that it is. The equation $x^2+y^2=0\tag{3}$ can be understood as describing the set of all points $(x,y,z)\in\Bbb R^3$ for which $x=0$ and $y=0$. In other words, it describes the $z$-axis.
So, we have described a line in $\Bbb R^3$ using a single scalar equation. But this means any line in $\Bbb R^3$ (or $\Bbb R^n$ for that matter) can be described by a single scalar equation, simply by using an appropriate affine transformation on the equation $(3)$.
As pointed out by Pantelis Damianou in the comment below, this gives us the equation $(x+y+z)^2+(3x+7y-1)^2=0$ in the case described above. Note that this tells us exactly the same thing as the equations $(1)$ and $(2)$, since $z^2+w^2=0$ is just another way to say that $z=0$ and $w=0$.
My question is:
Is this point of view ever useful? Does it have any striking applications? Is there an area of mathematics that uses such equations in a fruitful way?
Thanks.