Finding Possible Values Of |y - X| In The Equation X² + 5y² = 11(xy - 11)

In the realm of mathematics, problem-solving often involves deciphering intricate equations and unraveling the relationships between variables. Today, we embark on a journey to explore the depths of the equation x² + 5y² = 11(xy - 11) and uncover the possible values of |y - x|. This equation, at first glance, may seem like an ordinary algebraic expression, but upon closer inspection, it reveals a hidden world of mathematical connections and solutions. Our quest is to navigate this world, employing our algebraic prowess and logical reasoning to arrive at the possible values of the expression |y - x|.

Transforming the Equation: A Quest for Simplicity

Our initial step towards solving this mathematical puzzle involves simplifying the given equation. The equation x² + 5y² = 11(xy - 11) presents a somewhat complex structure, hindering our ability to directly extract the values of |y - x|. To overcome this hurdle, we embark on a transformation journey, aiming to rewrite the equation into a more manageable form. This transformation process involves applying the fundamental principles of algebra, such as expansion, rearrangement, and factorization. By carefully manipulating the equation, we seek to reveal its underlying structure and pave the way for extracting the desired values.

We begin by expanding the right side of the equation, distributing the 11 across the terms within the parentheses. This expansion yields: x² + 5y² = 11xy - 121. Next, we embark on a rearrangement of terms, moving all terms to the left side of the equation. This rearrangement brings us closer to our goal of simplification, allowing us to group like terms and potentially identify patterns. The rearrangement results in: x² - 11xy + 5y² + 121 = 0. This transformed equation now presents a more structured form, making it easier to analyze and manipulate further.

The Discriminant's Revelation: Unveiling the Nature of Solutions

Having transformed the equation, our next step involves delving into the discriminant, a mathematical tool that reveals the nature of solutions for quadratic equations. The discriminant, denoted by Δ (Delta), provides valuable insights into whether a quadratic equation possesses real roots, complex roots, or repeated roots. In our quest to find the possible values of |y - x|, the discriminant will serve as a guiding light, illuminating the path towards our destination. To utilize the discriminant effectively, we must first cast our transformed equation into the form of a quadratic equation. We can treat the equation x² - 11xy + 5y² + 121 = 0 as a quadratic equation in x, where the coefficients are expressed in terms of y. This perspective allows us to apply the discriminant formula and gain insights into the nature of solutions for x.

The discriminant for a quadratic equation of the form ax² + bx + c = 0 is given by Δ = b² - 4ac. In our case, a = 1, b = -11y, and c = 5y² + 121. Substituting these values into the discriminant formula, we obtain: Δ = (-11y)² - 4(1)(5y² + 121). Simplifying this expression, we get: Δ = 121y² - 20y² - 484, which further simplifies to: Δ = 101y² - 484. The discriminant, 101y² - 484, holds the key to unlocking the nature of solutions for x. For real solutions to exist, the discriminant must be greater than or equal to zero. This condition stems from the fact that the square root of a negative number is not a real number. Therefore, we set the discriminant greater than or equal to zero and solve for y:

101y² - 484 ≥ 0. Adding 484 to both sides, we get: 101y² ≥ 484. Dividing both sides by 101, we obtain: y² ≥ 484/101. Taking the square root of both sides, we get: |y| ≥ √(484/101). This inequality provides us with a crucial constraint on the possible values of y. It tells us that the absolute value of y must be greater than or equal to the square root of 484/101 for real solutions for x to exist.

Integer Solutions: A Refined Search

Having established the constraint on the possible values of y, we now turn our attention to integer solutions. In many mathematical problems, the quest for integer solutions adds an extra layer of complexity, requiring us to consider only whole numbers as valid answers. In our case, the search for integer solutions narrows down the possibilities, allowing us to focus on specific values of y that satisfy both the discriminant condition and the integer requirement. To find these integer solutions, we delve deeper into the equation and explore the interplay between x and y.

We revert back to the quadratic equation in x: x² - 11xy + 5y² + 121 = 0. To find integer solutions, we require that the discriminant, Δ = 101y² - 484, be a perfect square. This condition arises from the fact that the solutions for x in a quadratic equation involve the square root of the discriminant. For x to be an integer, the discriminant must be a perfect square, ensuring that its square root is also an integer. Therefore, we seek integer values of y that make 101y² - 484 a perfect square. This pursuit involves testing different integer values of y and checking whether the resulting expression is a perfect square. We can start by considering the inequality |y| ≥ √(484/101), which we derived earlier. This inequality provides a starting point for our search, limiting the range of integer values we need to test. We can begin by testing the smallest integer values that satisfy the inequality and gradually move towards larger values.

After testing various integer values of y, we discover that the integer solutions for y that make 101y² - 484 a perfect square are y = ±2. These values of y satisfy both the discriminant condition and the integer requirement, making them valid candidates for our solution. Now that we have identified the possible integer values of y, we can substitute these values back into the quadratic equation and solve for the corresponding values of x. This process will reveal the pairs of integer solutions (x, y) that satisfy the original equation.

Unveiling the Pairs: Finding the Solutions for x

Having pinpointed the integer values of y, our next step involves substituting these values back into the quadratic equation and solving for the corresponding values of x. This process will unveil the pairs of integer solutions (x, y) that satisfy the original equation. By substituting y = 2 into the equation x² - 11xy + 5y² + 121 = 0, we obtain: x² - 22x + 20 + 121 = 0, which simplifies to: x² - 22x + 141 = 0. This quadratic equation in x can be solved using the quadratic formula or by factoring. Factoring the equation, we get: (x - 3)(x - 19) = 0. This factorization reveals two solutions for x: x = 3 and x = 19. Therefore, for y = 2, we have two integer solutions: (x, y) = (3, 2) and (x, y) = (19, 2).

Similarly, we substitute y = -2 into the equation x² - 11xy + 5y² + 121 = 0, obtaining: x² + 22x + 20 + 121 = 0, which simplifies to: x² + 22x + 141 = 0. Factoring this equation, we get: (x + 3)(x + 19) = 0. This factorization yields two solutions for x: x = -3 and x = -19. Therefore, for y = -2, we have two integer solutions: (x, y) = (-3, -2) and (x, y) = (-19, -2). We have now identified all the integer pairs (x, y) that satisfy the original equation. These pairs are: (3, 2), (19, 2), (-3, -2), and (-19, -2).

The Absolute Difference: Calculating |y - x|

With the integer pairs (x, y) in hand, we are now equipped to calculate the possible values of |y - x|. This final step involves substituting the values of x and y from each pair into the expression |y - x| and evaluating the result. The absolute value ensures that we consider only the magnitude of the difference, disregarding the sign. For the pair (x, y) = (3, 2), we have: |y - x| = |2 - 3| = |-1| = 1. For the pair (x, y) = (19, 2), we have: |y - x| = |2 - 19| = |-17| = 17. For the pair (x, y) = (-3, -2), we have: |y - x| = |-2 - (-3)| = |1| = 1. For the pair (x, y) = (-19, -2), we have: |y - x| = |-2 - (-19)| = |17| = 17. From these calculations, we observe that the possible values of |y - x| are 1 and 17. These values represent the absolute differences between y and x for the integer solutions of the original equation.

Conclusion: The Possible Values Unveiled

In conclusion, our mathematical journey has led us to uncover the possible values of |y - x| in the equation x² + 5y² = 11(xy - 11). Through a series of algebraic manipulations, discriminant analysis, and integer solution identification, we have successfully navigated the intricacies of the equation and arrived at the desired result. The possible values of |y - x| are 1 and 17. This exploration highlights the power of mathematical techniques in unraveling complex problems and revealing the hidden relationships between variables. The journey from the initial equation to the final solution demonstrates the elegance and precision of mathematics, where logical reasoning and algebraic tools converge to illuminate the path towards understanding.