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Simplifying (4x + 3y + 2) * dx + (5x + 4y + 1) * dy = 0 Reorder the terms: (2 + 4x + 3y) * dx + (5x + 4y + 1) * dy = 0 Reorder the terms for easier multiplication: dx(2 + 4x + 3y) + (5x + 4y + 1) * dy = 0 (2 * dx + 4x * dx + 3y * dx) + (5x + 4y + 1) * dy = 0 Reorder the terms: (2dx + 3dxy + 4dx2) + (5x + 4y + 1) * dy = 0 (2dx + 3dxy + 4dx2) + (5x + 4y + 1) * dy = 0 Reorder the terms: 2dx + 3dxy + 4dx2 + (1 + 5x + 4y) * dy = 0 Reorder the terms for easier multiplication: 2dx + 3dxy + 4dx2 + dy(1 + 5x + 4y) = 0 2dx + 3dxy + 4dx2 + (1 * dy + 5x * dy + 4y * dy) = 0 Reorder the terms: 2dx + 3dxy + 4dx2 + (5dxy + 1dy + 4dy2) = 0 2dx + 3dxy + 4dx2 + (5dxy + 1dy + 4dy2) = 0 Reorder the terms: 2dx + 3dxy + 5dxy + 4dx2 + 1dy + 4dy2 = 0 Combine like terms: 3dxy + 5dxy = 8dxy 2dx + 8dxy + 4dx2 + 1dy + 4dy2 = 0 Solving 2dx + 8dxy + 4dx2 + 1dy + 4dy2 = 0 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Factor out the Greatest Common Factor (GCF), 'd'. d(2x + 8xy + 4x2 + y + 4y2) = 0Subproblem 1
Set the factor 'd' equal to zero and attempt to solve: Simplifying d = 0 Solving d = 0 Move all terms containing d to the left, all other terms to the right. Simplifying d = 0Subproblem 2
Set the factor '(2x + 8xy + 4x2 + y + 4y2)' equal to zero and attempt to solve: Simplifying 2x + 8xy + 4x2 + y + 4y2 = 0 Solving 2x + 8xy + 4x2 + y + 4y2 = 0 Move all terms containing d to the left, all other terms to the right. Add '-2x' to each side of the equation. 2x + 8xy + 4x2 + y + -2x + 4y2 = 0 + -2x Reorder the terms: 2x + -2x + 8xy + 4x2 + y + 4y2 = 0 + -2x Combine like terms: 2x + -2x = 0 0 + 8xy + 4x2 + y + 4y2 = 0 + -2x 8xy + 4x2 + y + 4y2 = 0 + -2x Remove the zero: 8xy + 4x2 + y + 4y2 = -2x Add '-8xy' to each side of the equation. 8xy + 4x2 + y + -8xy + 4y2 = -2x + -8xy Reorder the terms: 8xy + -8xy + 4x2 + y + 4y2 = -2x + -8xy Combine like terms: 8xy + -8xy = 0 0 + 4x2 + y + 4y2 = -2x + -8xy 4x2 + y + 4y2 = -2x + -8xy Add '-4x2' to each side of the equation. 4x2 + y + -4x2 + 4y2 = -2x + -8xy + -4x2 Reorder the terms: 4x2 + -4x2 + y + 4y2 = -2x + -8xy + -4x2 Combine like terms: 4x2 + -4x2 = 0 0 + y + 4y2 = -2x + -8xy + -4x2 y + 4y2 = -2x + -8xy + -4x2 Add '-1y' to each side of the equation. y + -1y + 4y2 = -2x + -8xy + -4x2 + -1y Combine like terms: y + -1y = 0 0 + 4y2 = -2x + -8xy + -4x2 + -1y 4y2 = -2x + -8xy + -4x2 + -1y Add '-4y2' to each side of the equation. 4y2 + -4y2 = -2x + -8xy + -4x2 + -1y + -4y2 Combine like terms: 4y2 + -4y2 = 0 0 = -2x + -8xy + -4x2 + -1y + -4y2 Simplifying 0 = -2x + -8xy + -4x2 + -1y + -4y2 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Solution
d = {0}
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