(x-2)dy+(y+4)dx=0

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Solution for (x-2)dy+(y+4)dx=0 equation: 


Simplifying
(x + -2) * dy + (y + 4) * dx = 0

Reorder the terms:
(-2 + x) * dy + (y + 4) * dx = 0

Reorder the terms for easier multiplication:
dy(-2 + x) + (y + 4) * dx = 0
(-2 * dy + x * dy) + (y + 4) * dx = 0

Reorder the terms:
(dxy + -2dy) + (y + 4) * dx = 0
(dxy + -2dy) + (y + 4) * dx = 0

Reorder the terms:
dxy + -2dy + (4 + y) * dx = 0

Reorder the terms for easier multiplication:
dxy + -2dy + dx(4 + y) = 0
dxy + -2dy + (4 * dx + y * dx) = 0
dxy + -2dy + (4dx + dxy) = 0

Reorder the terms:
4dx + dxy + dxy + -2dy = 0

Combine like terms: dxy + dxy = 2dxy
4dx + 2dxy + -2dy = 0

Solving
4dx + 2dxy + -2dy = 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(2x + xy + -1y) = 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 '(2x + xy + -1y)' equal to zero and attempt to solve:

Simplifying
2x + xy + -1y = 0

Solving
2x + xy + -1y = 0

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

Add '-2x' to each side of the equation.
2x + xy + -2x + -1y = 0 + -2x

Reorder the terms:
2x + -2x + xy + -1y = 0 + -2x

Combine like terms: 2x + -2x = 0
0 + xy + -1y = 0 + -2x
xy + -1y = 0 + -2x
Remove the zero:
xy + -1y = -2x

Add '-1xy' to each side of the equation.
xy + -1xy + -1y = -2x + -1xy

Combine like terms: xy + -1xy = 0
0 + -1y = -2x + -1xy
-1y = -2x + -1xy

Add 'y' to each side of the equation.
-1y + y = -2x + -1xy + y

Combine like terms: -1y + y = 0
0 = -2x + -1xy + y

Simplifying
0 = -2x + -1xy + y

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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