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Simplifying (2x + -1y) * dy = (4x2 + -4xy + y2 + 1) * dx Reorder the terms for easier multiplication: dy(2x + -1y) = (4x2 + -4xy + y2 + 1) * dx (2x * dy + -1y * dy) = (4x2 + -4xy + y2 + 1) * dx (2dxy + -1dy2) = (4x2 + -4xy + y2 + 1) * dx Reorder the terms: 2dxy + -1dy2 = (1 + -4xy + 4x2 + y2) * dx Reorder the terms for easier multiplication: 2dxy + -1dy2 = dx(1 + -4xy + 4x2 + y2) 2dxy + -1dy2 = (1 * dx + -4xy * dx + 4x2 * dx + y2 * dx) Reorder the terms: 2dxy + -1dy2 = (1dx + dxy2 + -4dx2y + 4dx3) 2dxy + -1dy2 = (1dx + dxy2 + -4dx2y + 4dx3) Solving 2dxy + -1dy2 = 1dx + dxy2 + -4dx2y + 4dx3 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1dx' to each side of the equation. 2dxy + -1dx + -1dy2 = 1dx + dxy2 + -4dx2y + -1dx + 4dx3 Reorder the terms: -1dx + 2dxy + -1dy2 = 1dx + dxy2 + -4dx2y + -1dx + 4dx3 Reorder the terms: -1dx + 2dxy + -1dy2 = 1dx + -1dx + dxy2 + -4dx2y + 4dx3 Combine like terms: 1dx + -1dx = 0 -1dx + 2dxy + -1dy2 = 0 + dxy2 + -4dx2y + 4dx3 -1dx + 2dxy + -1dy2 = dxy2 + -4dx2y + 4dx3 Add '-1dxy2' to each side of the equation. -1dx + 2dxy + -1dxy2 + -1dy2 = dxy2 + -4dx2y + -1dxy2 + 4dx3 Reorder the terms: -1dx + 2dxy + -1dxy2 + -1dy2 = dxy2 + -1dxy2 + -4dx2y + 4dx3 Combine like terms: dxy2 + -1dxy2 = 0 -1dx + 2dxy + -1dxy2 + -1dy2 = 0 + -4dx2y + 4dx3 -1dx + 2dxy + -1dxy2 + -1dy2 = -4dx2y + 4dx3 Add '4dx2y' to each side of the equation. -1dx + 2dxy + -1dxy2 + 4dx2y + -1dy2 = -4dx2y + 4dx2y + 4dx3 Combine like terms: -4dx2y + 4dx2y = 0 -1dx + 2dxy + -1dxy2 + 4dx2y + -1dy2 = 0 + 4dx3 -1dx + 2dxy + -1dxy2 + 4dx2y + -1dy2 = 4dx3 Add '-4dx3' to each side of the equation. -1dx + 2dxy + -1dxy2 + 4dx2y + -4dx3 + -1dy2 = 4dx3 + -4dx3 Combine like terms: 4dx3 + -4dx3 = 0 -1dx + 2dxy + -1dxy2 + 4dx2y + -4dx3 + -1dy2 = 0 Factor out the Greatest Common Factor (GCF), 'd'. d(-1x + 2xy + -1xy2 + 4x2y + -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 '(-1x + 2xy + -1xy2 + 4x2y + -4x3 + -1y2)' equal to zero and attempt to solve: Simplifying -1x + 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = 0 Solving -1x + 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = 0 Move all terms containing d to the left, all other terms to the right. Add 'x' to each side of the equation. -1x + 2xy + -1xy2 + 4x2y + -4x3 + x + -1y2 = 0 + x Reorder the terms: -1x + x + 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = 0 + x Combine like terms: -1x + x = 0 0 + 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = 0 + x 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = 0 + x Remove the zero: 2xy + -1xy2 + 4x2y + -4x3 + -1y2 = x Add '-2xy' to each side of the equation. 2xy + -1xy2 + 4x2y + -4x3 + -2xy + -1y2 = x + -2xy Reorder the terms: 2xy + -2xy + -1xy2 + 4x2y + -4x3 + -1y2 = x + -2xy Combine like terms: 2xy + -2xy = 0 0 + -1xy2 + 4x2y + -4x3 + -1y2 = x + -2xy -1xy2 + 4x2y + -4x3 + -1y2 = x + -2xy Add 'xy2' to each side of the equation. -1xy2 + 4x2y + -4x3 + xy2 + -1y2 = x + -2xy + xy2 Reorder the terms: -1xy2 + xy2 + 4x2y + -4x3 + -1y2 = x + -2xy + xy2 Combine like terms: -1xy2 + xy2 = 0 0 + 4x2y + -4x3 + -1y2 = x + -2xy + xy2 4x2y + -4x3 + -1y2 = x + -2xy + xy2 Add '-4x2y' to each side of the equation. 4x2y + -4x3 + -4x2y + -1y2 = x + -2xy + xy2 + -4x2y Reorder the terms: 4x2y + -4x2y + -4x3 + -1y2 = x + -2xy + xy2 + -4x2y Combine like terms: 4x2y + -4x2y = 0 0 + -4x3 + -1y2 = x + -2xy + xy2 + -4x2y -4x3 + -1y2 = x + -2xy + xy2 + -4x2y Add '4x3' to each side of the equation. -4x3 + 4x3 + -1y2 = x + -2xy + xy2 + -4x2y + 4x3 Combine like terms: -4x3 + 4x3 = 0 0 + -1y2 = x + -2xy + xy2 + -4x2y + 4x3 -1y2 = x + -2xy + xy2 + -4x2y + 4x3 Add 'y2' to each side of the equation. -1y2 + y2 = x + -2xy + xy2 + -4x2y + 4x3 + y2 Combine like terms: -1y2 + y2 = 0 0 = x + -2xy + xy2 + -4x2y + 4x3 + y2 Simplifying 0 = x + -2xy + xy2 + -4x2y + 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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