(2x+7y)dx+(7x+4y)dy=0

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Solution for (2x+7y)dx+(7x+4y)dy=0 equation: 


Simplifying
(2x + 7y) * dx + (7x + 4y) * dy = 0

Reorder the terms for easier multiplication:
dx(2x + 7y) + (7x + 4y) * dy = 0
(2x * dx + 7y * dx) + (7x + 4y) * dy = 0

Reorder the terms:
(7dxy + 2dx2) + (7x + 4y) * dy = 0
(7dxy + 2dx2) + (7x + 4y) * dy = 0

Reorder the terms for easier multiplication:
7dxy + 2dx2 + dy(7x + 4y) = 0
7dxy + 2dx2 + (7x * dy + 4y * dy) = 0
7dxy + 2dx2 + (7dxy + 4dy2) = 0

Reorder the terms:
7dxy + 7dxy + 2dx2 + 4dy2 = 0

Combine like terms: 7dxy + 7dxy = 14dxy
14dxy + 2dx2 + 4dy2 = 0

Solving
14dxy + 2dx2 + 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), '2d'.
2d(7xy + x2 + 2y2) = 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 '(7xy + x2 + 2y2)' equal to zero and attempt to solve:

Simplifying
7xy + x2 + 2y2 = 0

Solving
7xy + x2 + 2y2 = 0

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

Add '-7xy' to each side of the equation.
7xy + x2 + -7xy + 2y2 = 0 + -7xy

Reorder the terms:
7xy + -7xy + x2 + 2y2 = 0 + -7xy

Combine like terms: 7xy + -7xy = 0
0 + x2 + 2y2 = 0 + -7xy
x2 + 2y2 = 0 + -7xy
Remove the zero:
x2 + 2y2 = -7xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + 2y2 = -7xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + 2y2 = -7xy + -1x2
2y2 = -7xy + -1x2

Add '-2y2' to each side of the equation.
2y2 + -2y2 = -7xy + -1x2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
0 = -7xy + -1x2 + -2y2

Simplifying
0 = -7xy + -1x2 + -2y2

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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