Multiple choice

Solve the following equations: $(x^{2} - y^{2})(x - y) = 16xy, (x^{4} - y^{4}) (x^{2} - y^{2}) = 640 x^{2}y^{2}$

  1. $x=0, y=0$
  2. $x=9, y=3$
  3. $x=4, y=5$
  4. $x=3, y=9$
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D Correct answer
AI explanation

Let the given equations be (x^2 - y^2)(x - y) = 16xy and (x^4 - y^4)(x^2 - y^2) = 640x^2y^2, which can be rewritten as (x - y)^2(x + y) = 16xy and (x^2 - y^2)^2(x^2 + y^2) = 640x^2y^2. By dividing the second equation by the square of the first, we get (x^2 + y^2)/(x + y) = 2.5. Let the ratio y/x be r, which simplifies the relation to (1 + r^2)/(1 + r) = 2.5, or 2r^2 - 5r - 3 = 0. Factoring this quadratic gives (2r + 1)(r - 3) = 0, so r = 3 and y = 3x. Substituting y = 3x into the first original equation gives (x - 3x)^2(x + 3x) = 16x(3x), which simplifies to 4x^2(4x) = 48x^2, meaning x = 3. Consequently, y = 9, making the solution x = 3, y = 9.

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