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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.

Asaf Karagila
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user492494
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  • For lines, $ax+by+c=0$ is sometimes called "standard form", but in general "move everything to one side" is the tidiest description that comes to mind. – Mark S. Nov 28 '18 at 15:05

2 Answers2

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$$bxy+cy-ax=0$$

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 $x$.

  • Yves Daoust - Thank you for your helpful and relevant answer. I searched around using your terminology "implicit equation of a curve" and see that this is exactly what I was looking for. – user492494 Nov 28 '18 at 15:44
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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?

5xum
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  • While I appreciate your input, your answer doesn't relate to my actual question. My question was that the same equation in two different forms, one in the usual form of a function and the other where all terms have been moved over to one side of the equation, clearly have, well, two different forms. Is there a special name for the form that is equal to zero? – user492494 Nov 28 '18 at 15:21
  • Root is more often used for the solutions of univariate equations. –  Nov 28 '18 at 16:04