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For how many ordered pairs of positive integers \((x,y)\) does the equation \(\dfrac{1}{x} +\dfrac{2}{y}=\dfrac{1}{3}\) hold?

 Jan 22, 2019
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For how many ordered pairs of positive integers (x,y)  does the equation  \(\dfrac{1}{x} +\dfrac{2}{y}=\dfrac{1}{3}\\ \) hold?

 

\(\dfrac{1}{x} +\dfrac{2}{y}=\dfrac{1}{3}\\ \)

 

x=0 and y = zero are both asymptotes.

as x tends to infinity    y tends to 6

as y tends to infinity     x tends to 3

 

\(\dfrac{1}{x} +\dfrac{2}{y}=\dfrac{1}{3}\\ \frac{2}{y}=\frac{1}{3}-\frac{1}{x}\\ \frac{2}{y}=\frac{x-3}{3x}\\ \frac{y}{2}=\frac{3x}{x-3}\\ y=\frac{6x}{x-3}\\ \)

 

for y to be positive x>3

For x to be positive y>6

 

I found these points using  EXCEL.   I found 6 ordered pairs.

 

 Jan 22, 2019

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