Solve these simultaneous equations

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The discussion revolves around solving the simultaneous equations involving variables x and y. The initial approach leads to a fourth-degree polynomial, which simplifies to find values for x, specifically x=1 and x=6, though y cannot be determined for x=6. Participants suggest keeping fractions in their exact form for potential simplifications and propose alternative methods for solving the equations. One participant points out that there are additional solutions that may have been overlooked. The conversation emphasizes the need for a more efficient approach to solving the problem.
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Homework Statement
See attached.
Relevant Equations
Simultaneous equations
1693397130409.png
In my approach,
##\dfrac{x^2}{y}+ y =2.5x##

and we know that

##2y^2=9-x^2##

##y^2=\dfrac{9-x^2}{2}##

##(\dfrac{x^2}{y}+ y)(\dfrac{x^2}{y}+ y)=(2.5x)^2##

##\dfrac{x^4}{y^2} + x^2+x^2+y^2=6.25x^2##

##\dfrac{x^4}{y^2}+2x^2+\dfrac{9-x^2}{2}=6.25x^2##

##\dfrac{2x^4}{9-x^2} +2x^2+\dfrac{9-x^2}{2}=6.25x^2##

##4x^4+2x^2(18-2x^2)+(9-x^2)^2=6.25x^2(18-2x^2)##

...

##36x^2+81-18x^2+x^4=112x^2-12.5x^4##

##12.5x^4+x^4-94.5x^2+81=0##

##13.5x^4-94.5x^2+81=0##

##x_1=6## We cannot have ##y## values for ##x_1=6##

and for ##x_2=1 ⇒y= ±2##

There should be a better approach...will explore later.
 
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chwala said:
Homework Statement: See attached.
Relevant Equations: Simultaneous equations

View attachment 331231In my approach,
##\dfrac{x^2}{y}+ y =2.5x##

and we know that

##2y^2=9-x^2##

##y^2=\dfrac{9-x^2}{2}##

##(\dfrac{x^2}{y}+ y)(\dfrac{x^2}{y}+ y)=(2.5x)^2##

##\dfrac{x^4}{y^2} + x^2+x^2+y^2=6.25x^2##

##\dfrac{x^4}{y^2}+2x^2+\dfrac{9-x^2}{2}=6.25x^2##

##\dfrac{2x^4}{9-x^2} +2x^2+\dfrac{9-x^2}{2}=6.25x^2##

##4x^4+2x^2(18-2x^2)+(9-x^2)^2=6.25x^2(18-2x^2)##

...

##36x^2+81-18x^2+x^4=112x^2-12.5x^4##

##12.5x^4+x^4-94.5x^2+81=0##

##13.5x^4-94.5x^2+81=0##

##x_1=6## We cannot have ##y## values for ##x_1=6##

and for ##x_2=1 ⇒y= ±2##

There should be a better approach...will explore later.
I didn't look over the whole thing, but one suggestion: leave fractions as fractions (that 5/2 thing at the start.) Decimals are usually approximations... better to maintain the exact expressions. And who knows? Something might cancel later.

And your solution method is probably as good as it's going to get for this one.

-Dan

Addendum: I just looked at the graph. You are missing two solutions.
 
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topsquark said:
I didn't look over the whole thing, but one suggestion: leave fractions as fractions (that 5/2 thing at the start.) Decimals are usually approximations... better to maintain the exact expressions. And who knows? Something might cancel later.

And your solution method is probably as good as it's going to get for this one.

-Dan

Addendum: I just looked at the graph. You are missing two solutions.
Yes, i am aware that there are other solutions...which can easily be determined when solving for ##x##... i am seeking other better ways of solving this particular problem...
 
chwala said:
Homework Statement: See attached.
Relevant Equations: Simultaneous equations

1693397130409-png.png


There should be a better approach...will explore later.
Multiply both sides of the first equation, ##\displaystyle \frac x y +\frac y x = \frac 5 2 \ ## by ##\displaystyle 2xy## .

Rewrite as ##\displaystyle 2x^2 - 5xy + 2y^2 =0 ## .

Factor and solve for ##x## or for ##y##.

Plug the result into the other given equation and solve.
 
SammyS said:
Multiply both sides of the first equation, ##\displaystyle \frac x y +\frac y x = \frac 5 2 \ ## by ##\displaystyle 2xy## .

Rewrite as ##\displaystyle 2x^2 - 5xy + 2y^2 =0 ## .

Factor and solve for ##x## or for ##y##.

Plug the result into the other given equation and solve.
@SammyS thanks let me check this approach...
 
chwala said:
Homework Statement: See attached.
Relevant Equations: Simultaneous equations

1693397130409-png[1].png

In my approach,

. . .

##13.5x^4-94.5x^2+81=0##

##x_1=6## We cannot have ##y## values for ##x_1=6##

and for ##x_2=1 ⇒y= ±2##

There should be a better approach...will explore later.

Your workings are correct down to and including the degree 4 polynomial equation I have quoted.

Note that multiplying that equation by 2 gives:

## \displaystyle 27x^4-189x^2+162=0##

After factoring out 27 and discarding it, you get:

## \displaystyle x^4-7x^2+6=0## .

This factors easily into ## \displaystyle (x^4-1)(x^2-6)=0## , so that either ## \displaystyle x^2=1## or ## \displaystyle x^2=6## , rather than ## \displaystyle x=1## or ## \displaystyle x=6## as you had.

That still leaves you with some details to finish up to arrive at the overall solution.
 
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@SammyS yeah I will look at this again... will be back in a few days...
 
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