(4x+6y-3)dx+(6x-4y+1)dy=0

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Solution for (4x+6y-3)dx+(6x-4y+1)dy=0 equation: 


Simplifying
(4x + 6y + -3) * dx + (6x + -4y + 1) * dy = 0

Reorder the terms:
(-3 + 4x + 6y) * dx + (6x + -4y + 1) * dy = 0

Reorder the terms for easier multiplication:
dx(-3 + 4x + 6y) + (6x + -4y + 1) * dy = 0
(-3 * dx + 4x * dx + 6y * dx) + (6x + -4y + 1) * dy = 0

Reorder the terms:
(-3dx + 6dxy + 4dx2) + (6x + -4y + 1) * dy = 0
(-3dx + 6dxy + 4dx2) + (6x + -4y + 1) * dy = 0

Reorder the terms:
-3dx + 6dxy + 4dx2 + (1 + 6x + -4y) * dy = 0

Reorder the terms for easier multiplication:
-3dx + 6dxy + 4dx2 + dy(1 + 6x + -4y) = 0
-3dx + 6dxy + 4dx2 + (1 * dy + 6x * dy + -4y * dy) = 0

Reorder the terms:
-3dx + 6dxy + 4dx2 + (6dxy + 1dy + -4dy2) = 0
-3dx + 6dxy + 4dx2 + (6dxy + 1dy + -4dy2) = 0

Reorder the terms:
-3dx + 6dxy + 6dxy + 4dx2 + 1dy + -4dy2 = 0

Combine like terms: 6dxy + 6dxy = 12dxy
-3dx + 12dxy + 4dx2 + 1dy + -4dy2 = 0

Solving
-3dx + 12dxy + 4dx2 + 1dy + -4dy2 = 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(-3x + 12xy + 4x2 + y + -4y2) = 0


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 '(-3x + 12xy + 4x2 + y + -4y2)' equal to zero and attempt to solve:

Simplifying
-3x + 12xy + 4x2 + y + -4y2 = 0

Solving
-3x + 12xy + 4x2 + y + -4y2 = 0

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

Add '3x' to each side of the equation.
-3x + 12xy + 4x2 + y + 3x + -4y2 = 0 + 3x

Reorder the terms:
-3x + 3x + 12xy + 4x2 + y + -4y2 = 0 + 3x

Combine like terms: -3x + 3x = 0
0 + 12xy + 4x2 + y + -4y2 = 0 + 3x
12xy + 4x2 + y + -4y2 = 0 + 3x
Remove the zero:
12xy + 4x2 + y + -4y2 = 3x

Add '-12xy' to each side of the equation.
12xy + 4x2 + y + -12xy + -4y2 = 3x + -12xy

Reorder the terms:
12xy + -12xy + 4x2 + y + -4y2 = 3x + -12xy

Combine like terms: 12xy + -12xy = 0
0 + 4x2 + y + -4y2 = 3x + -12xy
4x2 + y + -4y2 = 3x + -12xy

Add '-4x2' to each side of the equation.
4x2 + y + -4x2 + -4y2 = 3x + -12xy + -4x2

Reorder the terms:
4x2 + -4x2 + y + -4y2 = 3x + -12xy + -4x2

Combine like terms: 4x2 + -4x2 = 0
0 + y + -4y2 = 3x + -12xy + -4x2
y + -4y2 = 3x + -12xy + -4x2

Add '-1y' to each side of the equation.
y + -1y + -4y2 = 3x + -12xy + -4x2 + -1y

Combine like terms: y + -1y = 0
0 + -4y2 = 3x + -12xy + -4x2 + -1y
-4y2 = 3x + -12xy + -4x2 + -1y

Add '4y2' to each side of the equation.
-4y2 + 4y2 = 3x + -12xy + -4x2 + -1y + 4y2

Combine like terms: -4y2 + 4y2 = 0
0 = 3x + -12xy + -4x2 + -1y + 4y2

Simplifying
0 = 3x + -12xy + -4x2 + -1y + 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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