Solving $bxy+cy-ax=0$ - NOT the roots

Question

I am looking for a bit of terminology but all I can ever find is explanations for finding the roots of equations, which is not what I am after.

Simply put, suppose we have a rational function, such as

$y=\displaystyle{\frac{ax}{bx+c}}$

and suppose we move everything over to one side of the equation, specifically as

$bxy+cy-ax=0$

so that we are essentially looking for all combinations of $x$ and $y$ such that the equation is zero. Does this form of the equation have a particular name? I thought "implicit" might work but that doesn't seem right. Homogeneous? Singular?

Thank you so much.

Answer 1

$$bxy+cy-ax=0$$

is the implicit equation of a curve and you are looking for the solution points $(x,y)$, or simply the solutions.

The form

$$y=\frac{ax}{bx+c}$$ is called explicit as it allows to directly compute $y$ knowing $x$.

Answer 2

The "combinations of $x$ and $y$ such that the equation is zero" makes no sense. An equation cannot be zero, just as an equation cannot be "elephant". An equation is either true or false.

And the phrase "combinations of $x$ and $y$ such that the equation $f(x,y)=0$ is true" does actually have a shorter term, and that is "the roots of $f$"

Why do you not want to use the term root?