Rearranging Equations: How to Express y as a Function of x?

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Discussion Overview

The discussion revolves around the rearrangement of equations to express one variable as a function of another, specifically focusing on the equation x² + 2xy - 3y² = 0. Participants explore methods for isolating y in terms of x, as well as the implications of doing so in terms of function definition. The conversation also touches on related equations and the challenges of finding inverses for higher-degree polynomials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express difficulty in rearranging equations with multiple instances of y, especially when they have different powers.
  • One participant suggests factoring the equation to find relationships between x and y, but another seeks to express y explicitly as a function of x.
  • There is a discussion about whether the original equation can be treated as a function, with some arguing it does not satisfy the definition of a function.
  • Several participants propose methods such as completing the square or using the quadratic formula to express y in terms of x, while noting that multiple values of y may exist for a single x.
  • Another participant raises a question about rearranging a different equation, Vi(t) + 1/2g(t)² = D, for t, leading to a discussion about quadratic equations and their solutions.
  • Some participants mention the inverse/implicit function theorems and the conditions under which local inverses may exist.
  • There is a mention of the limitations of solving polynomials of degree five or higher and the conditions under which certain polynomials can be solved.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the original equation can be expressed as a function of y. There are multiple competing views on how to approach the problem, and the discussion remains unresolved regarding the best methods for rearranging the equations.

Contextual Notes

Some participants note that the original equation does not define y as a function of x due to the potential for multiple outputs for a single input. Additionally, there are unresolved questions regarding the rearrangement of higher-degree polynomials and the existence of inverse functions.

Who May Find This Useful

This discussion may be useful for students or individuals interested in algebra, particularly those grappling with rearranging equations and understanding the implications of function definitions in mathematics.

beamthegreat
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I'm slightly embarrassed to ask such an easy question but how do you rearrange the following equation such that y=f(x)?

x2+2xy-3y2=0

I always have trouble with these types of problems when there are multiple "y"'s in an equation especially when they have different powers. Can someone please help?
 
Last edited:
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beamthegreat said:
I'm slightly embarrassed to ask such an easy question but how do you rearrange the following question such that y=f(x)?

x2+2xy-3y^2=0

I always have trouble with these types of problems when there are multiple "y"'s in an equation especially when they have different powers. Can someone please help?
The left side factors very easily. You will get two relationships with x as a function of y.
 
Can you clarify what you mean? If I factor the equation I get (x-y)(x+3y)=0 but I want it in a y=f(x) form.
 
It does not satisfy the definition of a function, why would you need to express it as one?
At any rate, I would go about expressing y by either completing the square or factorising and then expressing y.
 
Can you show how its done? I want to express it in a y=f(x) format since there is another function that intersects with this one.
Do you know any other way to find the coordinate that it intersects without having to express it in a y=f(x) format?
 
Yes, you have 2 canonic equations. You can express y or x from either one, plug it into the other equation and solve the resulting equation. Assuming those 2 sets intersect, the result will not be an empty set.
Again, the first is not a function, it's some sort of a relation between the two variables. It is a set, there can be intersection between sets.
 
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beamthegreat said:
I'm slightly embarrassed to ask such an easy question but how do you rearrange the following equation such that y=f(x)?

x2+2xy-3y2=0

I always have trouble with these types of problems when there are multiple "y"'s in an equation especially when they have different powers. Can someone please help?

First, consider x as fixed, then you have a quadratic equation in y. Simply solve that quadratic and you have y expressed in terms of x. If there is more than one value of y for a given x, then this will not be a function, but you could make it a function by restricting the values of y to be unique for each x.
 
PeroK said:
First, consider x as fixed, then you have a quadratic equation in y.

I'm not sure if I understand you. The problem still persist even if you try to rearrange the equation such that x=f(y).

Also, if you want to rearrange the equation Vi(t)+1/2g(t)^2=D for t such that t equals a function of Vi and g, how do you do it?

Can someone actually demonstrate it?
 
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beamthegreat said:
I'm not sure if I understand you. The problem still persist even if you try to rearrange the equation such that x=f(y).

Also, if you want to rearrange the equation Vi(t)+1/2g(t)^2=D for t such that t equals a function of Vi and g, how do you do it?

Can someone actually demonstrate it?
Your notation is confusing, especially Vi(t), which could be interpreted as Vi being a function of t, or V times i, which is a function of t.
I'm going to assume you mean this: vit + (1/2)gt2 = D. This equation is quadratic in t, so you can use the Quadratic Formula to solve for t. To make it a bit easier, write the equation in this form: gt2 + 2vit - D = 0.

In this form, a = g, b = 2vi, and c = -D. Just plug these into the Quadratic Formula to get t.
 
  • #10
It may be possible to rotate the plane to get rid of the mixed term.
 
  • #11
Mark44 said:
Your notation is confusing, especially Vi(t), which could be interpreted as Vi being a function of t, or V times i, which is a function of t.
Mark44 said:
I'm going to assume you mean this: vit + (1/2)gt2 = D. This equation is quadratic in t, so you can use the Quadratic Formula to solve for t. To make it a bit easier, write the equation in this form: gt2 + 2vit - D = 0.

In this form, a = g, b = 2vi, and c = -D. Just plug these into the Quadratic Formula to get t.

Alright, thanks.

What do do you do if you can't use the quadratic equation?

For example, how do you rearrange the following equation:

y=x^5+x^3+x^2+x+4 so that x=f(y)
 
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  • #12
beamthegreat said:
I'm not sure if I understand you. The problem still persist even if you try to rearrange the equation such that x=f(y).

Also, if you want to rearrange the equation Vi(t)+1/2g(t)^2=D for t such that t equals a function of Vi and g, how do you do it?

Can someone actually demonstrate it?

It's really not very difficult. ##x^2 +2xy - 3y^2 = 0 \ \ iff \ \ (x-y)(x+3y)=0 \ \ iff \ \ y = x \ \ or -\frac{x}{3}##

And that's y expressed in terms of x. As explained above, this is not a function of x.

If you want y = f(x), where (x, f(x)) is a solution to the equation, then you have two obvious options:

## y = f(x) = x## and ##y = f(x) = -\frac{x}{3}##
 
  • #13
beamthegreat said:
Alright, thanks.

What do do you do if you can't use the quadratic equation?

For example, how do you rearrange the following equation:

y=x^5+x^3+x^2+x+4 so that x=f(y)
I don't believe it's possible for this one. Did you make it up?
 
  • #14
Mark44 said:
I don't believe it's possible for this one. Did you make it up?

Yes I made it up. Is it really impossible to find the inverse of this equation?
 
  • #15
The inverse/implicit function theorems give conditions under which a local inverse (may)exist(s).
 
  • #16
WWGD said:
The inverse/implicit function theorems give conditions under which a local inverse (may)exist(s).

I see. What about other equations that has an inverse function?

For example, y=x^3+x^2 has an inverse function according to wolfram but how the heck do you solve it? Is there a rule/method you can use to solve these types of problems? What are these problems called anyway?
 
  • #17
There are formulas for solving quadratic, cubic, and quartic equations. It has been proven that there cannot be a formula, involving only roots, addition, multiplication, addition, and subtraction, for solving polynomials of degree 5 or higher.
 
  • #18
HallsofIvy said:
There are formulas for solving quadratic, cubic, and quartic equations. It has been proven that there cannot be a formula, involving only roots, addition, multiplication, addition, and subtraction, for solving polynomials of degree 5 or higher.

Do you mean for _all_ quintics (sorry if you do, I am no expert in this area)? I think if the Galois group of the polynomial is solvable, then you can use those operations only.
 
  • #19
I meant a general formula.. Yes, of course, there exist solutions to some polynomials of any degree.
 
  • #20
Ah, sorry, I am kind of new to that area.
 

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