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Hello, juriemagic here. Normally I do not have problems with solving simultaneous equations, but this one has got me a bit. I did work it out, but still feels un-easy about the answers. Would someone please help me with this one?:

 

Equation 1: x^2-y^2=0

Equation 2: x^2-4x-9=4y

 

All and any help will be greatly appreciated!!

Guest Nov 9, 2015

Best Answer 

 #2
avatar+91039 
+5

Hi Juriemagic,

It is good to see you again  laugh

 

Equation 1: \(x^2-y^2=0 \)

Equation 2: \(x^2-4x-9=4y\)

 

Mmm  This is an unusual one.

It always helps if you can picture what the graphs look like

 

\(x^2-y^2=0\\ x^2=y^2\\ y=\pm x\\ \mbox{This is a big X on the number plane centred on (0,0)}\\ \mbox{I can also see that the other one is a concave up parabola.}\\ \mbox{anyway that means}\\ x^2-4x-9=4x\qquad or \quad x^2-4x-9=-4x\\ x^2-8x-9=0 \qquad or \quad x^2-9=0\\ (x-9)(x+1)=0 \qquad or \quad (x-3)(x+3)=0\\ x=9\;\;\;or\;\;\;x=-1 \qquad or\qquad x=3\;\;\;or\;\;\;x=-3\\ \mbox{Now I am going to check if y should be pos or neg by subbing into equ2}\\ x=9 \qquad 81-36-9>0\;\;\;so\;\;\;y>0\;\;\;y=9\;\;\;(9,9)\\ x=-1 \qquad 1+4-9<0\;\;\;so\;\;\;y<0\;\;\;y=-1\;\;\;(-1,-1)\\ x=3 \qquad 9-12-9<0\;\;\;so\;\;\;y<0\;\;\;y=-3\;\;\;(3,-3)\\ x=-3 \qquad 9+12-9>0\;\;\;so\;\;\;y>0\;\;\;y=3\;\;\;(-3,3)\\ \)

 

So the 4 points of intersection (simuiltaneous solutions)  are  (9,9), (-1,-1),  (3,-3),  and   (-3,3)

 

Here is the graph to show you what is happening

https://www.desmos.com/calculator/lvjgdscib6

 

Melody  Nov 9, 2015
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6+0 Answers

 #1
avatar+18715 
+5

Equation 1: x^2-y^2=0

Equation 2: x^2-4x-9=4y

 

(1) (x-y)(x+y) = 0

   (x-y) = 0  \(\Rightarrow\)  x = y

   (x+y) = 0  \(\Rightarrow\) x = -y

  

\(y = \pm x\)

(2) x^2 -4x-9 = 4(-x)

     x^2  -9 = 0

     x^2 = 9

     \(x = \pm 3\)

 

    x^2 - 4x -9 = 4(x)

    x^2 - 8x - 9 = 0    factor

    (x+1)(x-9)= 0

    x+1 = 0   \(\Rightarrow\) \(x = -1\)

    x-9 = 0  \(\Rightarrow\)  \( x = 9\)

 

laugh

heureka  Nov 9, 2015
 #2
avatar+91039 
+5
Best Answer

Hi Juriemagic,

It is good to see you again  laugh

 

Equation 1: \(x^2-y^2=0 \)

Equation 2: \(x^2-4x-9=4y\)

 

Mmm  This is an unusual one.

It always helps if you can picture what the graphs look like

 

\(x^2-y^2=0\\ x^2=y^2\\ y=\pm x\\ \mbox{This is a big X on the number plane centred on (0,0)}\\ \mbox{I can also see that the other one is a concave up parabola.}\\ \mbox{anyway that means}\\ x^2-4x-9=4x\qquad or \quad x^2-4x-9=-4x\\ x^2-8x-9=0 \qquad or \quad x^2-9=0\\ (x-9)(x+1)=0 \qquad or \quad (x-3)(x+3)=0\\ x=9\;\;\;or\;\;\;x=-1 \qquad or\qquad x=3\;\;\;or\;\;\;x=-3\\ \mbox{Now I am going to check if y should be pos or neg by subbing into equ2}\\ x=9 \qquad 81-36-9>0\;\;\;so\;\;\;y>0\;\;\;y=9\;\;\;(9,9)\\ x=-1 \qquad 1+4-9<0\;\;\;so\;\;\;y<0\;\;\;y=-1\;\;\;(-1,-1)\\ x=3 \qquad 9-12-9<0\;\;\;so\;\;\;y<0\;\;\;y=-3\;\;\;(3,-3)\\ x=-3 \qquad 9+12-9>0\;\;\;so\;\;\;y>0\;\;\;y=3\;\;\;(-3,3)\\ \)

 

So the 4 points of intersection (simuiltaneous solutions)  are  (9,9), (-1,-1),  (3,-3),  and   (-3,3)

 

Here is the graph to show you what is happening

https://www.desmos.com/calculator/lvjgdscib6

 

Melody  Nov 9, 2015
 #3
avatar
+5

My goodness!!!,

 

Melody and Heureka!!, you did it again!.. smiley. It's soooo different from how I did it!. The X^2=y^2 is really what had me. Awww!, and it's really so simple!. Thank you to both of you!, you are really superb!!.

Guest Nov 9, 2015
 #4
avatar+91039 
0

Thank you for all the flattery Juriemagic :))    laugh

 

You need to get used to using Desmos Graphing calculator (or any graphing calc) to help you understand what is happening with these problems.

Learning to anticipate what a wide variety of graphs will look like will help you with many problems. 

 

I see that you are logged on at present.  I hoope this means that you log on problems are over  frown

Melody  Nov 9, 2015
 #5
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0

hi Melody, you mean you can see I'm logged on?...strange, because the reply still gives me as "guest", and I also still get the "cookie" error. ??sad 

Guest Nov 9, 2015
 #6
avatar+91039 
0

Hi Juriemagic,

Yes earlier your name was displayed as being logged on.   ://    That is an enigma.   crying sad sad

I hope this problems is fixed soon. ://

Melody  Nov 9, 2015

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