January 6th, 2019, 12:30 PM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  Am I right to assume this?
Hi everyone. So, I've been working on a problem, and I ended up with the equation $$1=(3x+y)(x+3y)$$ where $x$ and $y$ are positive real numbers. So, I found that the solution is $(\frac{1}{4},\frac{1}{4})$ because the only way this can be true is if each factor is $1$. Am I correct that this should be the only solution? I'm basically asking, is it okay to just say, we get $(\frac{1}{4},\frac{1}{4})$ since this is only true if each factor is $1$. Last edited by ProofOfALifetime; January 6th, 2019 at 12:53 PM. 
January 6th, 2019, 12:38 PM  #2 
Senior Member Joined: Dec 2015 From: Earth Posts: 327 Thanks: 42 
Equation has infinite pairs (x,y) $\displaystyle (\frac{1}{4} , \frac{1}{4} )$ is a solution , but not the only one 
January 6th, 2019, 12:50 PM  #3 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research! 
Okay thank you! That makes sense.

January 6th, 2019, 01:00 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 
$1 = 3x^2 + 10xy + 3y^2$ $3x^2 + 10 xy + 3y^2 1 = 0$ $x = \dfrac{10y \pm \sqrt{100y^2 12(3y^21)}}{6}$ $x = \dfrac{10y\pm\sqrt{64y^2+12}}{6}$ $x = \dfrac{5y\pm \sqrt{16y^2+3}}{3}$ 
January 6th, 2019, 01:08 PM  #5 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research! 
Thank you! I guess I will keep playing around with the original problem. You both are a big help!

January 6th, 2019, 01:13 PM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,561 Thanks: 2562 Math Focus: Mainly analysis and algebra  Quote:
 
January 6th, 2019, 01:24 PM  #7 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  
January 6th, 2019, 01:26 PM  #8  
Senior Member Joined: May 2016 From: USA Posts: 1,252 Thanks: 519  Quote:
$3x + y = a \ne 0 \text { and } x + 3y = \dfrac{1}{a} \implies (3x + y)(x + 3y) = 1.$ Last edited by JeffM1; January 6th, 2019 at 01:51 PM.  
January 6th, 2019, 01:37 PM  #9 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  Exactly why I asked the question! I was thinking the same thing, but since I work on these challenge problems alone, I like to get advice from people with more knowledge and experience.
Last edited by skipjack; January 6th, 2019 at 02:33 PM. 
January 6th, 2019, 02:54 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,105 Thanks: 1907 
What was the original problem that you were working on?


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