Question 445209
.
i'm trying to show a=b or a=1/b without plugging in the values for a.
this is the equation
a+1/a=b+1/b
i have tried multiplying both sides by ab
i tried the quadratic equation but that didn't work very well cause i couldn't get the square root out
i just need a step by step explanation of how to get there cause its driving me crazy and i think i'm just missing something simple
please remember. i know how to substitute. i'm basically proving a= b,1/b
which is why i tried making it a quadratic because i knew there were 2 answers
thank you very much
i hope you can help
~~~~~~~~~~~~~~~~~~~~~~~~~



Good question and good problem.


@mananth was on a right track, but his solution was incomplete.
I came to present a complete solution/answer/exposition/explanation.



<pre>
    a + {{{1/a}}} = b + {{{1/b}}}

    {{{(a^2+1)/a}}} = {{{(b^2+1)/b)}}}

multiply by ab

    {{{b*(a^2+1)}}} = {{{a*(b^2+1)}}}

    {{{b*a^2 + b}}} = {{{ab^2+a}}}

    {{{ba^2 = ab^2}}} = a - b

    ab(a-b)=(a-b)

    ab(a-b) - (a-b) = 0

    (ab-1)*(a-b) = 0


So, EITHER ab = 1,  which is the same as  a = {{{1/b}}}  (one of the two your options) 

    OR     a = b   (second of the two your options).
</pre>

Solved completely.


My congrats !