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

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


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

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

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

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

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

Combine like terms: 6dxy + 6dxy = 12dxy
12dxy + 6dx2 + 8dy2 = 0

Solving
12dxy + 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(6xy + 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 '(6xy + 3x2 + 4y2)' equal to zero and attempt to solve:

Simplifying
6xy + 3x2 + 4y2 = 0

Solving
6xy + 3x2 + 4y2 = 0

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

Add '-6xy' to each side of the equation.
6xy + 3x2 + -6xy + 4y2 = 0 + -6xy

Reorder the terms:
6xy + -6xy + 3x2 + 4y2 = 0 + -6xy

Combine like terms: 6xy + -6xy = 0
0 + 3x2 + 4y2 = 0 + -6xy
3x2 + 4y2 = 0 + -6xy
Remove the zero:
3x2 + 4y2 = -6xy

Add '-3x2' to each side of the equation.
3x2 + -3x2 + 4y2 = -6xy + -3x2

Combine like terms: 3x2 + -3x2 = 0
0 + 4y2 = -6xy + -3x2
4y2 = -6xy + -3x2

Add '-4y2' to each side of the equation.
4y2 + -4y2 = -6xy + -3x2 + -4y2

Combine like terms: 4y2 + -4y2 = 0
0 = -6xy + -3x2 + -4y2

Simplifying
0 = -6xy + -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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