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how do you solve this ?? please help me 2x^2-3y^2=5xy and-3x+y=5

Guest Jan 9, 2015

Best Answer 

 #5
avatar+90988 
+10

Thank you Chris for discovering this for us  

 

This Wikipaedia site that Chris has referred us too is interesting.

http://www.digplanet.com/wiki/Degenerate_conic

 

Unlike Chris I like to put my feet up and watch video clips.  There are tabs at the the top and one is for youtube clips.

I just watched the first one and really liked it.

 

--------------------------------------------

Actually I just found another simple page one conics.  

My knowledge of conics was worse than Chris's in the first place so this is quite enlightening.

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

This is what I have learned:

 

The general form of a conic is    

$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0\\

Now if B=0 we have\\

Ax^2+Cy^2+Dx+Ey+F=0\\

If A=C it is a circle\\

If A\ne C \;\; \mbox{BUT A and C have the same sign then it is an ellipse}\\

If A\ne C \;\; \mbox{AND A and C have different signs then it is an hyperbola}\\$$

---------------------------------------------------------

Ours has an xy term so B is not equal to zero - I do not know what that makes it.

It is a degenerate conic that is for sure but i am not sure which one.     

 

I would really like more imput from other mathematicians  :)

Melody  Jan 9, 2015
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6+0 Answers

 #1
avatar+6 
0

it = 4 7

distinysucks  Jan 9, 2015
 #2
avatar+78577 
+10

2x^2-3y^2=5xy and-3x+y=5     rearranging the second equation, we have y = 3x + 5

And putting this into the first equation, we have

2x^2 - 3(3x + 5)^2 = 5x(3x + 5)  simplify

2x^2 - 3(9x^2 + 30x +25) = 15x^2 + 25x

2x^2 -27x^2 - 90x - 75  = 15x^2 + 25x

-13x^2 - 27x^2 - 115x - 75  = 0   multiply through by -1

13x^2 + 27x^2  + 115x + 75 = 0

40x^2 + 115x + 75 = 0      divide through by 5

8x^2 + 23x + 15 = 0    factor

(x + 1 )(8x + 15) = 0

And setting each factor to 0, we have that x = -1 and x = -15/8

And using y = 3x + 5 when x = -1, y = 2  and when x = -15/8, y = -5/8

So....our solutions are (-1, 2) and (-15/8, -5/8)

Here's a graph.......https://www.desmos.com/calculator/mdhdt8ytqq

The "blue" function is a straight line.....the other  one is something I haven't seen before....a pair of intersecting lines that are "rotated"....very odd !!!!

 --------------------------------------------------------------------------------------------------------------

P.S. - I found an internet source that says the strange graph is known as a "degenerate" conic....!!!!

CPhill  Jan 9, 2015
 #3
avatar+90988 
+5

This is the most unexpected graph ever.  Mmm one for the interest posts definitely !

 Thanks for bringing it to my attention Chris   

Melody  Jan 9, 2015
 #4
avatar+78577 
+10

A little more on this strange graph.....

Apparently, if this can be factored into this form....(x + y) (x - y) = 0, we have a graph of intersecting lines...let's see...

2x^2 - 3y^2 = 5xy 

2x^2 - 5xy - 3y^2 = 0

(2x + y) (x - 3y)  = 0

Since this "reducible" to this form, this is a degenerate conic that will form two intersecting lines.

Here's the graph, again.....https://www.desmos.com/calculator/hwmmfcew1u

Notice something......if we set the first term in the above factorization to 0, we have 2x + y = 0, or just y = -2x...and this is the line ine on the graph that "falls" from right to left!!!  Similarly, doing the same thing to the second factored term produces y = (1/3)x.....and this is the other line on the graph that "rises" from left to right....!!!

And notice one last thing......just like we might do in a quadratic by "factoring and setting to 0" to find the roots....we are doing something similar here....except that, instead of generating "roots," we're generating equations of lines....!!!!

Here's some more info about these odd graphs....http://www.digplanet.com/wiki/Degenerate_conic

 

CPhill  Jan 9, 2015
 #5
avatar+90988 
+10
Best Answer

Thank you Chris for discovering this for us  

 

This Wikipaedia site that Chris has referred us too is interesting.

http://www.digplanet.com/wiki/Degenerate_conic

 

Unlike Chris I like to put my feet up and watch video clips.  There are tabs at the the top and one is for youtube clips.

I just watched the first one and really liked it.

 

--------------------------------------------

Actually I just found another simple page one conics.  

My knowledge of conics was worse than Chris's in the first place so this is quite enlightening.

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

This is what I have learned:

 

The general form of a conic is    

$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0\\

Now if B=0 we have\\

Ax^2+Cy^2+Dx+Ey+F=0\\

If A=C it is a circle\\

If A\ne C \;\; \mbox{BUT A and C have the same sign then it is an ellipse}\\

If A\ne C \;\; \mbox{AND A and C have different signs then it is an hyperbola}\\$$

---------------------------------------------------------

Ours has an xy term so B is not equal to zero - I do not know what that makes it.

It is a degenerate conic that is for sure but i am not sure which one.     

 

I would really like more imput from other mathematicians  :)

Melody  Jan 9, 2015
 #6
avatar+752 
0

2x^2 - 3y^2 = 5xy

2x^2 -5xy -3y^2 = 0

2x^2 -6xy +1xy -3y^2 =0

2x(x -3y) +y(x-3y) =0

(x-3y)(2x +y) =0

x-3y = 0 or 2x+y =0

x =3y or 2x =-y

              x= -y/2

Sasini  Jan 9, 2015

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