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Simplifying (3x + 5y) * dx + (4x + 6y) * dy = 0 Reorder the terms for easier multiplication: dx(3x + 5y) + (4x + 6y) * dy = 0 (3x * dx + 5y * dx) + (4x + 6y) * dy = 0 Reorder the terms: (5dxy + 3dx2) + (4x + 6y) * dy = 0 (5dxy + 3dx2) + (4x + 6y) * dy = 0 Reorder the terms for easier multiplication: 5dxy + 3dx2 + dy(4x + 6y) = 0 5dxy + 3dx2 + (4x * dy + 6y * dy) = 0 5dxy + 3dx2 + (4dxy + 6dy2) = 0 Reorder the terms: 5dxy + 4dxy + 3dx2 + 6dy2 = 0 Combine like terms: 5dxy + 4dxy = 9dxy 9dxy + 3dx2 + 6dy2 = 0 Solving 9dxy + 3dx2 + 6dy2 = 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), '3d'. 3d(3xy + x2 + 2y2) = 0 Factor a trinomial. 3d((x + y)(x + 2y)) = 0 Ignore the factor 3.Subproblem 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 '(x + y)' equal to zero and attempt to solve: Simplifying x + y = 0 Solving x + y = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x' to each side of the equation. x + -1x + y = 0 + -1x Combine like terms: x + -1x = 0 0 + y = 0 + -1x y = 0 + -1x Remove the zero: y = -1x Add '-1y' to each side of the equation. y + -1y = -1x + -1y Combine like terms: y + -1y = 0 0 = -1x + -1y Simplifying 0 = -1x + -1y The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 3
Set the factor '(x + 2y)' equal to zero and attempt to solve: Simplifying x + 2y = 0 Solving x + 2y = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x' to each side of the equation. x + -1x + 2y = 0 + -1x Combine like terms: x + -1x = 0 0 + 2y = 0 + -1x 2y = 0 + -1x Remove the zero: 2y = -1x Add '-2y' to each side of the equation. 2y + -2y = -1x + -2y Combine like terms: 2y + -2y = 0 0 = -1x + -2y Simplifying 0 = -1x + -2y 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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