X = ((384x/(y+384))*y+384(384x/(y+384)))/384

[johnnyamerica]

Homework Statement

These are different way's I've phrased the question:

x = ((384x/(y+384))*y+384(384x/(y+384)))/384

x = (1280((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))))/(-sqrt(768^2-((768^2-(((-1280)(-y))/(-y-384))^2)))+1280)

x = ((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y+384-y))

v= 384x/(y+384)
v= (sqrt(768^2-(((-1280)(-y))/(-y-384))^2))
y= (384x-384v)/v

v= 768cos((pi((180t)/pi))/180)
t= acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768)


The original version:
x=
(1280(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi
))/180)))/(-sqrt(768^2-(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi))/180))^2)+1280)


How do I solve for y, even if there's multiple values for y?

 

[NascentOxygen]

I'm having trouble figuring out what this is about. Are all of the sets of equations below supposed to have an identical solution y=f(x)?

These are different way's I've phrased the question:

x = ((384x/(y+384))*y+384(384x/(y+384)))/384

This reduces to 0=0

v= 768cos((pi((180t)/pi))/180)
t= acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768)

Are you wanting to solve for y in terms of v here?

Is acos your abbreviation for arccos?

How do I solve for y, even if there's multiple values for y?

You know there are multiple solutions?

 

[johnnyamerica]

Affirmative. Every equation I've written as x= gives an identical answer for a given y. If y=216 x=960.

960 = (((384*960)/(216+384))*216+384((384*960)/(216+384)))/384

960 = (1280((sqrt(768^2-(((-1280)(-216))/(-216-384))^2))))/(-sqrt(768^2-((768^2-(((-1280)(-216))/(-216-384))^2)))+1280)

960 = ((sqrt(768^2-(((-1280)(-216))/(-216-384))^2)))((216+384)/(216+384-216))

960 = (1280(768cos((pi((180acos((sqrt(768^2-(((-1280)(-216))/(-216-384))^2))/768))/pi
))/180)))/(-sqrt(768^2-(768cos((pi((180acos((sqrt(768^2-(((-1280)(-216))/(-216-384))^2))/768))/pi))/180))^2)+1280)


However, I need to rewrite these to find y if the /only information i have/ is x and the rest of the equation, or rather with no knowledge of y.

y=0, x=768:

768 = ((sqrt(768^2-(((-1280)(-0))/(-0-384))^2)))((0+384)/(0+384-0))


The v and t examples are just more alternative ways of writing the same equation. If I could somehow convert any given x into v or t- with no knowledge of y- I could solve this equation for y. But my goal is to solve for y so that I can find y for any given x, not just x=960 or x=768.

 

[Borek]

If your equation reduces to 0=0, it is an identity, so in a way every pair x,y is a solution.

 

[johnnyamerica]

Nascent's wrong, it isn't 0=0.

It's ( y(384x/(y+384)) + 384(384x/(y+384)) )/384 =x

So how do I find y=...?

 

[Borek]

Discussing with facts is not the best way of solving problems:

http://www.wolframalpha.com/input/?i=x-+((384x/(y+384))*y+384(384x/(y+384)))/384

 

[johnnyamerica]

x-x=0, you say? I wasn't aware.

Any value of x or y can result in a different value of x or y. I can find any x if I know y, but how can I find any y without already knowing y?

 

[Office_Shredder]

The point is ANY value of y will work. It doesn't even matter what value of x you've picked. The only exception to this is if y=-384 people will complain about the denominator being 0

 

[D H]

Homework Statement

These are different way's I've phrased the question:

x = ((384x/(y+384))*y+384(384x/(y+384)))/384

x = (1280((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))))/(-sqrt(768^2-((768^2-(((-1280)(-y))/(-y-384))^2)))+1280)

x = ((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y+384-y))

Re the first expression:
This is almost a tautology. (It is a tautology except at y=-384, where it is undefined.)Re the second expression:
This is not at all the same as your first expression. You can reduce this expression further. Those (((-1280)(-y))/(-y-384))² terms are better written as 1280²(y/(y+384))². You also have a bunch of superfluous parentheses.

It will help to create auxiliary variables w=x/256 and z=y/(y+384) and to write 768 as 256*3 and 1280 as 256*5. Note that common factor of 256.Re the third expression:
This is the same as your second expression when y≥0. It is not the same as your second expression when y is negative. Just as you can reduce the second expression further, you can also reduce this third expression considerably. For example, y+384-y is just 384. Here it also helps to recognize that 384=256*3/2 (that common factor of 256 appears yet again).

 

[berkeman]

x-x=0, you say? I wasn't aware.

Are you being sincere or sarcastic here? That would seem to answer your question, no?

 

[johnnyamerica]

I get that. I just want to solve this to equal y, so that if I know what x is I can solve for y.

If I know that y=216, I know that x=960. But what if I knew that x=961, let's say? I'd have zero clue what y=, and I'd have to guess.

I tried to find x=y, and by guestimation the closet I came was 518.883098582601474166600894~=x=y but my calculator won't go farther.

It's a curve, x changes with y. Like x^2=y, x= either sqrt(y) or -sqrt(y)

y=576, x=0. y=0, x=768. y=216, x=960.

y=-144, x=0

It may help to know that between y=576 and y=-144 is the area I'm interested in.

 

[D H]

If you know that y is non-negative, I'd go with your third expression. If y can be positive or negative, you'll need to use a simplified version of your second expression.

 

[johnnyamerica]

Are you being sincere or sarcastic here? That would seem to answer your question, no?

If x=y, x-x=0. If x=(e^(it) + e^(-it))/2, x-x=0. If cos(x)=5, x-x=0.
If x=0, x-x=0.

 

[SammyS]

Homework Statement

These are different way's I've phrased the question:

(1) x = ((384x/(y+384))*y+384(384x/(y+384)))/384

(2) x = (1280((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))))/(-sqrt(768^2-((768^2-(((-1280)(-y))/(-y-384))^2)))+1280)

(3) x = ((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y+384-y))

v= 384x/(y+384)
v= (sqrt(768^2-(((-1280)(-y))/(-y-384))^2))
y= (384x-384v)/v

v= 768cos((pi((180t)/pi))/180)
t= acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768)

The original version:
(4) x=
(1280(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi
))/180)))/(-sqrt(768^2-(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi))/180))^2)+1280)

How do I solve for y, even if there's multiple values for y?

I fed these expressions to WolframAlpha.

The one I labeled (1) above is equivalent to x = x. So, x can be any number, y can be any number except -384 .

According to WolframAlpha, the others are all equivalent to \displaystyle x=\frac{8}{3} \sqrt{-(y-576) (y+144)}\,, provided that y > 0. Of course this expression
is real provided that -144 ≤ y ≤ 576 .

Also, the first expression is the only one with x on the left hand side (as well as being on the right).

 

[johnnyamerica]

I need to solve for y. Without already knowing y.
All these equations depend on already knowing what y is.

If x=2y. y=x/2.
If x=(e^(iy) + e^(-iy))/2, y= either -ilog(x-sqrt(x^2-1)) or -ilog(x+sqrt(x^2-1))

If x= ((384x/(y+384))*y+384(384x/(y+384)))/384

y= ?

 

[SammyS]

...

If x= ((384x/(y+384))*y+384(384x/(y+384)))/384

y= ?

Now I understand what you're asking.

It's been mentioned by some, if not all, of the responders to this thread, that any value of y (except that y≠-384, because that would give division by zero) will satisfy this equation no matter what the value of x.

To spell it out, the right-hand side of your expression can be simplified as follows:

\displaystyle\frac{1}{384}\left(\frac{384xy}{y+384}+\frac{384^2x}{y+384}\right)

\displaystyle=\left(\frac{xy}{y+384}+\frac{384x}{y+384}\right)

\displaystyle=\left(\frac{xy+384x}{y+384}\right)

\displaystyle=x\left(\frac{y+384}{y+384}\right)
​

 

[johnnyamerica]

If x= ((384x/(y+384))*y+384(384x/(y+384)))/384
y= ?

If y^2=x, y must either be
y1= sqrt(x), y2= -sqrt(x)

If ((384x/(y+384))*y+384(384x/(y+384)))/384 = x = 0

Then y1=576 and y2=-144

It's a curve. There are two values of y for any x except at the cusp of the curve.

How can I take a value of x and calculate the two values of y without knowing either value of y?

 

[D H]

What exactly are you trying to solve?

Your tautology is not a curve. You made a mistake somewhere in deriving x= ((384x/(y+384))*y+384(384x/(y+384)))/384.

 

[johnnyamerica]

((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y+384-y)) = x

If you swap x and y and type y=((sqrt(768^2-(((-1280)(-x))/(-x-384))^2)))((x+384)/(x+384-x)) into even the simplest graphing calculator you can see the curve.

I'm trying to solve y=?

Or x in the calculator-friendly version...


I get what everyone is saying: that the equations I've provided are all identities because they are all true. But they all have y lodged in them somewhere. How do I isolate y so that I can solve for y without knowing y.

An equation that says whatever(x)=y.

Two equations for y, one that turns x=0 into y1=-144 and one that turns x=0 into y2=576

 

[NascentOxygen]

This might be a good time to take a breather.

How about describing where this equation/formula came from, for starters. Maybe you are on the wrong track all together, and even were you able to 'solve' this it might be wasted effort if you've run off-course even before reaching this point.

You seem to have put in a lot of effort already; but let's back up a way and confirm that you are on track to solving something tangible.

 

[NascentOxygen]

((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y[/color]+384-y[/color])) = x

The mathematical term for what you appear to be wanting to do is to make y the subject of the formula.

However, before anyone plunges into helping with this, you need to double- and triple- check your equation here. I have highlighted two y-terms which appear to cancel. This doesn't inspire confidence that the equation you present is an accurate portrayal of what you are trying to solve.

 

[johnnyamerica]

The mathematical term for what you appear to be wanting to do is to make y the subject of the formula.

However, before anyone plunges into helping with this, you need to double- and triple- check your equation here. I have highlighted two y-terms which appear to cancel. This doesn't inspire confidence that the equation you present is an accurate portrayal of what you are trying to solve.

Lol, yea, that tends to happen. I found four redundancies in the original equation. I've tested it repeatedly and for every y between -144 and 576 it's correct.

x=((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(384))


----
v= 384x/(y+384)
v= (sqrt(768^2-(((-1280)(-y))/(-y-384))^2))
y= (384x-384v)/v

v= 768cos((pi((180t)/pi))/180)
t= acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768)
---

If I could somehow get x to equal v or t I can easily solve this for y. But these are just examples because everyone approaches problems differently.

 

[D H]

However, before anyone plunges into helping with this, you need to double- and triple- check your equation here. I have highlighted two y-terms which appear to cancel. This doesn't inspire confidence that the equation you present is an accurate portrayal of what you are trying to solve.

I pointed that out in post #9.

The equation that johnnyamerica has has been using in the past few posts is the one that SammyS labeled as equation (3) in post #14. It follows from equation (2) assuming y is non-negative. Equation (2) in turn follows from equation (4).

All three of those equations are a mess. They simplify immensely. Equation (3) simplifies to

x=\frac 8 3 \sqrt{(576-y)(144+y)}

I'd suggest using the substitution z=y/(384+y). This works very nicely with equation (2).

Lol, yea, that tends to happen. I found four redundancies in the original equation. I've tested it repeatedly and for every y between -144 and 576 it's correct.

It only tends to happen when you are being sloppy. Sorry for being blunt, but a bit of bluntness is needed here.

Your first equation,

x = ((384x/(y+384))*y+384(384x/(y+384)))/384

is not the same as the other equations. You made a mistake somewhere. Since you haven't shown your work, there's no telling where that mistake arose.


Your original equation is

x=
(1280(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi
))/180)))/(-sqrt(768^2-(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi))/180))^2)+1280)

This is an absolute mess. You should get rid of things like (-1280)(-y) ASAP. It is just 1280y. Similarly, (-y-384)^2 is just (y+384)^2.


Your second equation is

x = (1280((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))))/(-sqrt(768^2-((768^2-(((-1280)(-y))/(-y-384))^2)))+1280)

This is indeed equivalent to your original equation. It is a lot simpler, but it is still a mess.


Your third equation is

x = ((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((y+384)/(y+384-y))

This is the same as equations (2) and (4) if y is non-negative. This is not a valid reduction in the case that y is negative.


You need to be a bit more careful.

 

[johnnyamerica]

They're a mess because they're the result of weeks of work. Sorry to be blunt but ALL of the equations in my first post are correct on both sides of the equation. They're given just as different possible starting locations in case someone dislikes one of them.

((384x/(y+384))*y+384(384x/(y+384)))/384=x
((384(960)/((216)+384))*(216)+384(384(960)/((216)+384)))/384=960
((384(0)/((576)+384))*(576)+384(384(0)/((576)+384)))/384=0
((384(0)/((-144)+384))*(-144)+384(384(0)/((-144)+384)))/384=0

x=
(1280(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi
))/180)))/(-sqrt(768^2-(768cos((pi((180acos((sqrt(768^2-(((-1280)(-y))/(-y-384))^2))/768))/pi))/180))^2)+1280)

x = ((sqrt(768^2-(((-1280)(-y))/(-y-384))^2)))((|y+384|)/(|y+384-y|))

Sorry, I changed this last one from absolute values because the symbol would not pass my calculator. However for the range of y=576 and y=-144 it works regardless.

This is a simple curve, I just cannot figure out what step to take to solve this for y, or make y the subject of the function.

y1=-144, y2=576, x=0
y1=0, y2=432, x=768
I'm estimating it peaks at y1=y2=216, x=960

 

[D H]

They're a mess because they're the result of weeks of work. Sorry to be blunt but ALL of the equations in my first post are correct on both sides of the equation.

No, they are not. That first one is a tautology. It reduces to x=x so long as y≠384. The second and third are only equivalent for y≥0.

Your third equation simplifies to

x=\frac 8 3 \sqrt{(576-y)(144+y)}

Solving for y given x is just a quadratic equation. Square both sides.

 

[johnnyamerica]

I don't know what a tautology is or what it is about this equation that makes it impossible to find y, but for any x other than 960 for which there can only be one value of y, there are two values of y - between -144 and 576.

I'm guessing making y the subject of these functions is beyond the skill of anyone here but I do appreciate you all telling me a million times that X is INDEED equal to X, and that SUBTRACTING SOMETHING FROM ITSELF MAKES THIS SOMETHING NO LONGER ANYTHING.

 

[D H]

Sigh. A tautology is something like x=x. That is exactly what your first equation reduces to. What that means is that given some x, any value of y except 384 will satisfy that first equation.

 

[johnnyamerica]

Sigh. A tautology is something like x=x. That is exactly what your first equation reduces to. What that means is that given some x, any value of y except 384 will satisfy that first equation.

Any value of y will satisfy that X is indeed equal to X? Oh, thank you.

So, like if (e^(iy) + e^(-iy))/2=x, and y were 5, let's say, then x=x?

Next you'll tell me that x-x=0. I'd require proof, however.

 

[NascentOxygen]

D.H. has given you the formula your complicated one/s simplify to. If you square both sides you end up with a quadratic equation in y. Are you able to solve that, to get y in terms of x^2?

It's not that no one is prepared to help you. But if you were able to come up with all those versions of the equation, then it's likely you can solve the quadratic in y. If you need further help, just ask.

 

[NascentOxygen]

General Q to others: anyone know how to get wolframalpha to change the subject of the formula?

 

[johnnyamerica]

Solved: x=((384x/(y+384))*y+384(384x/(y+384)))/384

I solved for y1 and y2. I can't edit the title of the original post but this is solved.

 

[SammyS]

General Q to others: anyone know how to get wolframalpha to change the subject of the formula?

@ NascentOxygen,

Do you mean like this?

http://www.wolframalpha.com/input/?i=solve+{x%3D+%28%28384x%2F%28y%2B384%29%29*y%2B384%28384x%2F%28y%2B384%29%29%29%2F384}+for+y

or this ?

http://www.wolframalpha.com/input/?i=solve+y+%3D+%285x%2B2%29%2F%283x-7%29+for+x

 

[NascentOxygen]

@ NascentOxygen,

Do you mean like this?

Precisely. Would be invaluable for checking results. Thanks.

 

[Borek]

x-x=0, you say? I wasn't aware.

Let's start with something simpler. Like x=3y-4. Easy to solve. Try to feed x-3y+4 to Wolfram and see if it will equal zero. Do you see there is some important difference between x = ((384x/(y+384))*y+384(384x/(y+384)))/384 and x=3y-4? One is tautology, other is not.

Any value of x or y can result in a different value of x or y. I can find any x if I know y, but how can I find any y without already knowing y?

Using your equation you can't find y, because such a unique solution doesn't exist.

x-x=y-y is another equation similar to yours - it can't be solved for exactly the same reasons you can't solve your equation. No matter what you put as x, any y will still make it correct. Yours is just more convoluted.