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Simplifying (x + -1y)(4x + y) * dx + (5x + -1y) * dy = 0 Reorder the terms for easier multiplication: dx(x + -1y)(4x + y) + (5x + -1y) * dy = 0 Multiply (x + -1y) * (4x + y) dx(x(4x + y) + -1y * (4x + y)) + (5x + -1y) * dy = 0 dx((4x * x + y * x) + -1y * (4x + y)) + (5x + -1y) * dy = 0 Reorder the terms: dx((xy + 4x2) + -1y * (4x + y)) + (5x + -1y) * dy = 0 dx((xy + 4x2) + -1y * (4x + y)) + (5x + -1y) * dy = 0 dx(xy + 4x2 + (4x * -1y + y * -1y)) + (5x + -1y) * dy = 0 dx(xy + 4x2 + (-4xy + -1y2)) + (5x + -1y) * dy = 0 Reorder the terms: dx(xy + -4xy + 4x2 + -1y2) + (5x + -1y) * dy = 0 Combine like terms: xy + -4xy = -3xy dx(-3xy + 4x2 + -1y2) + (5x + -1y) * dy = 0 (-3xy * dx + 4x2 * dx + -1y2 * dx) + (5x + -1y) * dy = 0 Reorder the terms: (-1dxy2 + -3dx2y + 4dx3) + (5x + -1y) * dy = 0 (-1dxy2 + -3dx2y + 4dx3) + (5x + -1y) * dy = 0 Reorder the terms for easier multiplication: -1dxy2 + -3dx2y + 4dx3 + dy(5x + -1y) = 0 -1dxy2 + -3dx2y + 4dx3 + (5x * dy + -1y * dy) = 0 -1dxy2 + -3dx2y + 4dx3 + (5dxy + -1dy2) = 0 Reorder the terms: 5dxy + -1dxy2 + -3dx2y + 4dx3 + -1dy2 = 0 Solving 5dxy + -1dxy2 + -3dx2y + 4dx3 + -1dy2 = 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(5xy + -1xy2 + -3x2y + 4x3 + -1y2) = 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 '(5xy + -1xy2 + -3x2y + 4x3 + -1y2)' equal to zero and attempt to solve: Simplifying 5xy + -1xy2 + -3x2y + 4x3 + -1y2 = 0 Solving 5xy + -1xy2 + -3x2y + 4x3 + -1y2 = 0 Move all terms containing d to the left, all other terms to the right. Add '-5xy' to each side of the equation. 5xy + -1xy2 + -3x2y + 4x3 + -5xy + -1y2 = 0 + -5xy Reorder the terms: 5xy + -5xy + -1xy2 + -3x2y + 4x3 + -1y2 = 0 + -5xy Combine like terms: 5xy + -5xy = 0 0 + -1xy2 + -3x2y + 4x3 + -1y2 = 0 + -5xy -1xy2 + -3x2y + 4x3 + -1y2 = 0 + -5xy Remove the zero: -1xy2 + -3x2y + 4x3 + -1y2 = -5xy Add 'xy2' to each side of the equation. -1xy2 + -3x2y + 4x3 + xy2 + -1y2 = -5xy + xy2 Reorder the terms: -1xy2 + xy2 + -3x2y + 4x3 + -1y2 = -5xy + xy2 Combine like terms: -1xy2 + xy2 = 0 0 + -3x2y + 4x3 + -1y2 = -5xy + xy2 -3x2y + 4x3 + -1y2 = -5xy + xy2 Add '3x2y' to each side of the equation. -3x2y + 4x3 + 3x2y + -1y2 = -5xy + xy2 + 3x2y Reorder the terms: -3x2y + 3x2y + 4x3 + -1y2 = -5xy + xy2 + 3x2y Combine like terms: -3x2y + 3x2y = 0 0 + 4x3 + -1y2 = -5xy + xy2 + 3x2y 4x3 + -1y2 = -5xy + xy2 + 3x2y Add '-4x3' to each side of the equation. 4x3 + -4x3 + -1y2 = -5xy + xy2 + 3x2y + -4x3 Combine like terms: 4x3 + -4x3 = 0 0 + -1y2 = -5xy + xy2 + 3x2y + -4x3 -1y2 = -5xy + xy2 + 3x2y + -4x3 Add 'y2' to each side of the equation. -1y2 + y2 = -5xy + xy2 + 3x2y + -4x3 + y2 Combine like terms: -1y2 + y2 = 0 0 = -5xy + xy2 + 3x2y + -4x3 + y2 Simplifying 0 = -5xy + xy2 + 3x2y + -4x3 + y2 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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