This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Express y in terms of x from the first equation. Given the equation xy = 1, we can write y as: y = (1)/(x) This assumes x 0. Step 2: Substitute the expression for y into the second equation. The second equation is t^2 x - y = t^3 - (1)/(t). Substitute y = (1)/(x): t^2 x - (1)/(x) = t^3 - (1)/(t) This assumes t 0. Step 3: Eliminate the denominators by multiplying by xt. Multiply both sides of the equation by xt: xt (t^2 x - (1)/(x)) = xt (t^3 - (1)/(t)) t^3 x^2 - t = t^4 x - x Step 4: Rearrange the equation into a quadratic form in terms of x. Move all terms to one side to form a quadratic equation Ax^2 + Bx + C = 0: t^3 x^2 - t^4 x + x - t = 0 t^3 x^2 - (t^4 - 1)x - t = 0 Step 5: Solve the quadratic equation for x using the quadratic formula. The quadratic formula is x = -B ± sqrt(B^2 - 4AC)2A. Here, A = t^3, B = -(t^4 - 1), and C = -t. x = -(-(t^4 - 1)) ± sqrt((-(t^4 - 1))^2 - 4(t^3)(-t))2(t^3) x = t^4 - 1 ± sqrt((t^4 - 1)^2 + 4t^4)2t^3 x = t^4 - 1 ± sqrt(t^8 - 2t^4 + 1 + 4t^4)2t^3 x = t^4 - 1 ± sqrt(t^8 + 2t^4 + 1)2t^3 Recognize that t^8 + 2t^4 + 1 = (t^4 + 1)^2: x = t^4 - 1 ± sqrt((t^4 + 1)^2)2t^3 x = (t^4 - 1 ± (t^4 + 1))/(2t^3) Step 6: Calculate the two possible values for x. For the positive root: x_1 = ((t^4 - 1) + (t^4 + 1))/(2t^3) = (2t^4)/(2t^3) = t For the negative root: x_2 = ((t^4 - 1) - (t^4 + 1))/(2t^3) = (t^4 - 1 - t^4 - 1)/(2t^3) = (-2)/(2t^3) = -(1)/(t^3) Step 7: Find the corresponding values for y using y = (1)/(x). For x_1 = t: y_1 = (1)/(t) For x_2 = -(1)/(t^3): y_2 = (1)/(-1)t^3 = -t^3 The solutions for the system of equations are: x=t, y=(1)/(t) or x=-(1)/(t^3), y=-t^3 What's next? Send 'em!