There is a curiosity that you may encounter whereby we have an equation in variables $x$ and $y$ and when it is graphed, it appears to be two or more straight lines. Indeed it could appear to be a circle and line, or even two circles, but we will confine this discussion to lines and you may explore and extrapolate on your own. Suppose we have the equation of two lines. For example: $L_{1}:\;y=2x-3$ $L_{2}:\;y=2x+1$ These two equations can multiplied together and expressed as $(y-2x+3)(y-2x-1)=$
$4\;x^{2}-4\;x\;y-4\;x+y^{2}+2\;y=3$
To graph the equation we would probably do it “implicitly”. That means, we would supply some $x$ values and compute $y$ values, building a table of points which would then be graphed. We call it an implicit plot because we did not explicitly define a function in one of the variables. This is very common whenever there exists both $x^{2}$ and $y^{2}$ terms and ever more so when there are mixed, $xy$, terms as well. These can appear to be very complicated expressions, but their independent graph serves as a clue that they can be factored.

In the figure, we show the equation
$ \left( -24x^{3}+26x^{2}y+120x^{2}+\right.$
$ 17xy^{2}-87.5xy-199.5x- $
$ \left. 12y^{2}+73.5y=-110.25 \right)$
which graphs as three lines because it can be factored into
$\small \mathbf{(4x+3y-7)(-3x+4y+5.25)(2x-y-3)=0.}$