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# help

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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

#1
+101745
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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