Solve for x and y: 1/x - 1/y = 1/3, 1/x2 + 1/y2 = 5/9.
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Answer / Rohit Rathi
First, we can rewrite the first equation as xy = 3. To eliminate the squares in the second equation, let's square both sides: (1/x)² + (1/y)² = 5/9 → x² + y² - 6xy/3 = 5/9 * 9 = 45. Now we have a quadratic equation in terms of xy: x² + y² - 6xy/3 = 45. Using the rational root theorem, we can test possible rational roots by plugging them into our quadratic equation. After some trial and error, we find that x = 1 is a solution because (1)² + y² - 6(1)(y)/3 = 45 becomes y² - 6y + 45 = 0, which factors as (y-5)(y-9)=0. So y = 5 or y = 9 are the possible values for y when x=1. Now we can find the corresponding values for x: When y=5, 1/x - 1/5 = 1/3 implies x=3, and when y=9, 1/x - 1/9 = 1/3 implies x=6. We have found two pairs of solutions (x,y) = (3,5), (6,9).
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