Solve for ##u## and ##v## in the given equations

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The discussion revolves around solving the equations involving variables u and v, specifically ##u-v=\dfrac{1}{6(u+v)}## and ##\dfrac{1}{u+v} + 12(u+v)=8##. Two values for ##u+v##, denoted as m, are derived: ##m_1=\dfrac{1}{2}## and ##m_2=\dfrac{1}{6}##, leading to simultaneous equations for each case. The solutions yield two pairs: ##(u,v) = \left(\frac{7}{12}, -\frac{5}{12}\right)## and ##(u,v) = \left(\frac{5}{12}, \frac{1}{12}\right)##. The discussion also touches on alternative approaches and potential errors in calculations, emphasizing the complexity of the problem.
chwala
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Homework Statement
See attached.
Relevant Equations
Understanding of simultaneous equations
1693395062637.png


In my approach i have:

##u-v=\dfrac{1}{6(u+v)}##

##\dfrac{1}{u+v} + 12(u+v)=8##

##1+12(u+v)^2=8(u+v)##

Let

##u+v=m##

then we shall have,

##12m^2-8m+1=0##

##m_1=\dfrac{1}{2}## and ##m_2=\dfrac{1}{6}##

Using ##m_2=\dfrac{1}{6}## and considering
##(u+v)(u-v)=\dfrac{1}{6}##
then,
##\dfrac{1}{6} (u-v)=\dfrac{1}{6}##

then we shall have the simultaneous equation,

##u-v=1##
##u+v=\dfrac{1}{6}## giving us
##u=\dfrac{7}{12} ⇒v=-\dfrac{5}{12}##

also using

##m_1=\dfrac{1}{2}##
then we shall have the simultaneous equation,
##u-v=\dfrac{1}{3}##
##u+v=\dfrac{1}{2}## giving us
##u=\dfrac{5}{12} ⇒v=\dfrac{1}{12}##

There may be another approach hence my post. Cheers.
 
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chwala said:
Homework Statement: See attached.
Relevant Equations: Understanding of simultaneous equations

There may be another approach
I would have started ##\frac 1{u+v}+\frac 2{u-v}=\frac{u-v+2(u+v)}{u^2-v^2}=(3u+v)6##.
No idea whether that is better.
 
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haruspex said:
I would have started ##\frac 1{u+v}+\frac 2{u-v}=\frac{u-v+2(u+v)}{u^2-v^2}=(3u+v)6##.
No idea whether that is better.
@haruspex let me check that out...
 
The other approach to this question though tedious

From,
##(u+v)(u-v)=\dfrac{1}{6}##

and noting that,

##3u-v=\dfrac{4}{3}## from the first equation then,

##(u+\dfrac{4}{3}-3u)(u+\dfrac{4}{3}+3u)=\dfrac{1}{6}##

...

##\dfrac{72u-16-72u^2}{9}=\dfrac{1}{6}##

##-432u^2+432u-96=9##

##432u^2-432u+105=0##

##144u^2-144u+35=0##

##u =\dfrac{144±\sqrt {20736-20160}}{288} ##

##= \dfrac{144±24}{288}##

##u_1=\dfrac{7}{12} ⇒v_1 = \dfrac{-5}{12}##

##u_2=\dfrac{5}{12} ⇒v_1 = \dfrac{1}{12}##
 
You are given ##\frac 1{u+v} + \frac 2{u-v} = 8## and ##(u+v)(u-v) = \frac 16##. An apprach is...

Let ##x = u+v## and ##y = u-v##. Also maybe use symbols for the constants: with ##a=2, b=8,## and ##c = \frac 16## the equations become:

##\frac 1x + \frac ay =b## and ##xy = c##.

From the first one ##y + ax = bxy##. Since ##xy = c## this becomes ##y + ax = bc##.

Substituting ##y = bc~ – ~ax## into ##xy = c## gives a quadratic in ##x## and the rest should follow.

Replace ##a, b## and ##c## when convenient.

Edit: typo's fixed.
 
Last edited:
With x = 1/(u + v), y = 2/(u-v) we have <br /> x + y = 8 \quad xy = 12 so that x and y are the roots of <br /> z^2 - 8z + 3 = (z - 4)^2 - 4 = 0 giving z = 4 \pm 2 and hence \begin{split}<br /> (u,v) &amp;= \left( \frac12\left(\frac1x + \frac2y\right), \frac12\left(\frac1x - \frac2y\right)\right) \\<br /> &amp;= \left( \frac 7{12}, -\frac{5}{12} \right) \quad\mbox{or}\quad \left( \frac{5}{12}, -\frac{1}{12}\right)\end{split}
 
pasmith said:
With x = 1/(u + v), y = 2/(u-v) we have <br /> x + y = 8 \quad xy = 12 so that x and y are the roots of <br /> z^2 - 8z + 3 = (z - 4)^2 - 4 = 0
I think there's a typo' and the last equation should be
##z^2 - 8z + 12 = (z - 4)^2 - 4 = 0##

pasmith said:
giving z = 4 \pm 2 and hence \begin{split}<br /> (u,v) &amp;= \left( \frac12\left(\frac1x + \frac2y\right), \frac12\left(\frac1x - \frac2y\right)\right) \\<br /> &amp;= \left( \frac 7{12}, -\frac{5}{12} \right) \quad\mbox{or}\quad \left( \frac{5}{12}, -\frac{1}{12}\right)\end{split}
Maybe I’ve made a mistake but it seems ##(u,v) = (\frac 5{12}, -\frac 1{12})## is not a solution...

##u+v = \frac 5{12} + \frac {-1}{12} = \frac 4{12} = \frac 13##

##u-v = \frac 5{12} - \frac {-1}{12} = \frac 6{12} = \frac 12##

##\frac 1{u+v}+ \frac 2{u-v} = \frac 1{\frac 13} + \frac 2{\frac 12} = 3+ 4 =7## (but it should be 8)

EDIT. I'm guessing it's just a sign-error so ##(u,v) = (\frac 5{12}, -\frac 1{12})## should be ##(u,v) = (\frac 5{12}, \frac 1{12})##.
 
Last edited:
pasmith said:
With x = 1/(u + v), y = 2/(u-v) we have <br /> x + y = 8 \quad xy = 12 so that x and y are the roots of <br /> z^2 - 8z + 3 = (z - 4)^2 - 4 = 0 giving z = 4 \pm 2
Minor typo.

##\displaystyle z^2 - 8z + 3 = 0 \ ## should be ##\displaystyle z^2 - 8z + 12 = 0 \ ##

The rest follows from that without change.
 
SammyS said:
Minor typo.

##\displaystyle z^2 - 8z + 3 = 0 \ ## should be ##\displaystyle z^2 - 8z + 12 = 0 \ ##

The rest follows from that without change.
Except that there is also the sign error in post #6 that @Steve4Physics noted in post #7.
 
  • #10
chwala said:
In my approach i have:

##u-v=\dfrac{1}{6(u+v)}##

##\dfrac{1}{u+v} + 12(u+v)=8##

##1+12(u+v)^2=8(u+v)##

Let

##u+v=m##

then we shall have,

##12m^2-8m+1=0##

##m_1=\dfrac{1}{2}## and ##m_2=\dfrac{1}{6}##

Using ##m_2=\dfrac{1}{6}## and considering
##(u+v)(u-v)=\dfrac{1}{6}##
then,
##\dfrac{1}{6} (u-v)=\dfrac{1}{6}##

then we shall have the simultaneous equation,

##u-v=1##
##u+v=\dfrac{1}{6}## giving us
##u=\dfrac{7}{12} ⇒v=-\dfrac{5}{12}##

also using

##m_1=\dfrac{1}{2}##
then we shall have the simultaneous equation,
##u-v=\dfrac{1}{3}##
##u+v=\dfrac{1}{2}## giving us
##u=\dfrac{5}{12} ⇒v=\dfrac{1}{12}##
Good work.
 
  • #11
Defining ##x=u+v##, ##y=u-v## the second equation is ##xy=\frac 16##, or ##6x=1/y##.

Substituting into the first gives ##\frac 1{x}+12x=8##, which rearranges to ##12x^2-8x+1=0##. Hence ##x=1/2## or ##x=1/6##, with the corresponding ##y=1/3## or ##y=1##.

Since ##u## and ##v## are obviously ##\frac 12(x\pm y)##, then we have ##(u,v)=\left(\frac{5}{12},\frac{1}{12}\right)## or ##\left(\frac{7}{12},\frac{-5}{12}\right)##.
 
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