(6x+4y)dx+(4x+8y)dy=0

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Solution for (6x+4y)dx+(4x+8y)dy=0 equation: 


Simplifying
(6x + 4y) * dx + (4x + 8y) * dy = 0

Reorder the terms for easier multiplication:
dx(6x + 4y) + (4x + 8y) * dy = 0
(6x * dx + 4y * dx) + (4x + 8y) * dy = 0

Reorder the terms:
(4dxy + 6dx2) + (4x + 8y) * dy = 0
(4dxy + 6dx2) + (4x + 8y) * dy = 0

Reorder the terms for easier multiplication:
4dxy + 6dx2 + dy(4x + 8y) = 0
4dxy + 6dx2 + (4x * dy + 8y * dy) = 0
4dxy + 6dx2 + (4dxy + 8dy2) = 0

Reorder the terms:
4dxy + 4dxy + 6dx2 + 8dy2 = 0

Combine like terms: 4dxy + 4dxy = 8dxy
8dxy + 6dx2 + 8dy2 = 0

Solving
8dxy + 6dx2 + 8dy2 = 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), '2d'.
2d(4xy + 3x2 + 4y2) = 0

Ignore the factor 2.

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 = 0

Subproblem 2
Set the factor '(4xy + 3x2 + 4y2)' equal to zero and attempt to solve:

Simplifying
4xy + 3x2 + 4y2 = 0

Solving
4xy + 3x2 + 4y2 = 0

Move all terms containing d to the left, all other terms to the right.

Add '-4xy' to each side of the equation.
4xy + 3x2 + -4xy + 4y2 = 0 + -4xy

Reorder the terms:
4xy + -4xy + 3x2 + 4y2 = 0 + -4xy

Combine like terms: 4xy + -4xy = 0
0 + 3x2 + 4y2 = 0 + -4xy
3x2 + 4y2 = 0 + -4xy
Remove the zero:
3x2 + 4y2 = -4xy

Add '-3x2' to each side of the equation.
3x2 + -3x2 + 4y2 = -4xy + -3x2

Combine like terms: 3x2 + -3x2 = 0
0 + 4y2 = -4xy + -3x2
4y2 = -4xy + -3x2

Add '-4y2' to each side of the equation.
4y2 + -4y2 = -4xy + -3x2 + -4y2

Combine like terms: 4y2 + -4y2 = 0
0 = -4xy + -3x2 + -4y2

Simplifying
0 = -4xy + -3x2 + -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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