Solving $bxy+cy-ax=0$ - NOT the roots
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3
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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.
terminology
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
asked Nov 28 at 15:02
user492494
285
add a comment |
up vote
3
down vote
favorite
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.
terminology
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
asked Nov 28 at 15:02
user492494
285
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 at 15:05
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
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.
terminology
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
asked Nov 28 at 15:02
user492494
285
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.
terminology
terminology
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
asked Nov 28 at 15:02
user492494
285
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
asked Nov 28 at 15:02
user492494
285
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
edited Nov 28 at 17:40
Asaf Karagila♦
300k32421751
300k32421751
asked Nov 28 at 15:02
user492494
285
asked Nov 28 at 15:02
user492494
285
asked Nov 28 at 15:02
user492494
285
285
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 at 15:05
add a comment |
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 at 15:05
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 at 15:05
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 at 15:05
add a comment |
2 Answers
2
active
oldest
votes
up vote
7
down vote
accepted
$$bxy+cy-ax=0$$
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 $x$.
answered Nov 28 at 15:11
Yves Daoust
122k668218
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 at 15:44
add a comment |
up vote
2
down vote
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?
answered Nov 28 at 15:06
5xum
88.9k393160
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
$$bxy+cy-ax=0$$
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 $x$.
answered Nov 28 at 15:11
Yves Daoust
122k668218
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 at 15:44
add a comment |
up vote
7
down vote
accepted
$$bxy+cy-ax=0$$
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 $x$.
answered Nov 28 at 15:11
Yves Daoust
122k668218
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 at 15:44
add a comment |
up vote
7
down vote
accepted
up vote
7
down vote
accepted
$$bxy+cy-ax=0$$
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 $x$.
answered Nov 28 at 15:11
Yves Daoust
122k668218
$$bxy+cy-ax=0$$
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 $x$.
answered Nov 28 at 15:11
Yves Daoust
122k668218
edited Nov 28 at 17:19
answered Nov 28 at 15:11
Yves Daoust
122k668218
answered Nov 28 at 15:11
Yves Daoust
122k668218
answered Nov 28 at 15:11
Yves Daoust
122k668218
122k668218
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 at 15:44
add a comment |
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 at 15:44
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 at 15:44
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 at 15:44
add a comment |
up vote
2
down vote
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?
answered Nov 28 at 15:06
5xum
88.9k393160
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
add a comment |
up vote
2
down vote
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?
answered Nov 28 at 15:06
5xum
88.9k393160
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
add a comment |
up vote
2
down vote
up vote
2
down vote
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?
answered Nov 28 at 15:06
5xum
88.9k393160
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?
answered Nov 28 at 15:06
5xum
88.9k393160
answered Nov 28 at 15:06
5xum
88.9k393160
answered Nov 28 at 15:06
5xum
88.9k393160
answered Nov 28 at 15:06
5xum
88.9k393160
88.9k393160
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
add a comment |
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
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 at 15:21
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 at 15:21
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
Root is more often used for the solutions of univariate equations.
– Yves Daoust
Nov 28 at 16:04
add a comment |
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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 at 15:05