Are there anymore ??

Algebra Level 3

1 a − b = 1 a × b \begin{aligned} \dfrac{1}{a-b } & = \dfrac{1}{a\times b}\end{aligned} The above relation holds true for a = 1 a=1 and b = 1 2 b=\frac{1}{2} . Are there any other non zero real numbers a a and b b such that above expression is true?

Inspiration

No Yes

Naren Bhandari by Naren Bhandari , 21, India

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2 solutions

Edwin Gray
Aug 13, 2018

Assume neither a nor b can equal 0, or the right side is indeterminate. Otherwise, let a - b = ab, or a= b + ab = b(1 + a), and b = a/(a +1). So there are infinitely many solutions. Ed Gray

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Michael Mendrin
Aug 12, 2018

Infinitely many, such as a = 2 , b = 2 3 a=2, b=\dfrac{2}{3}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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