(6x+7y)dx+(7x+4y)dy=0

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Solution for (6x+7y)dx+(7x+4y)dy=0 equation: 


Simplifying
(6x + 7y) * dx + (7x + 4y) * dy = 0

Reorder the terms for easier multiplication:
dx(6x + 7y) + (7x + 4y) * dy = 0
(6x * dx + 7y * dx) + (7x + 4y) * dy = 0

Reorder the terms:
(7dxy + 6dx2) + (7x + 4y) * dy = 0
(7dxy + 6dx2) + (7x + 4y) * dy = 0

Reorder the terms for easier multiplication:
7dxy + 6dx2 + dy(7x + 4y) = 0
7dxy + 6dx2 + (7x * dy + 4y * dy) = 0
7dxy + 6dx2 + (7dxy + 4dy2) = 0

Reorder the terms:
7dxy + 7dxy + 6dx2 + 4dy2 = 0

Combine like terms: 7dxy + 7dxy = 14dxy
14dxy + 6dx2 + 4dy2 = 0

Solving
14dxy + 6dx2 + 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 + 3x2 + 2y2) = 0

Factor a trinomial.
2d((3x + y)(x + 2y)) = 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 '(3x + y)' equal to zero and attempt to solve:

Simplifying
3x + y = 0

Solving
3x + y = 0

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

Add '-3x' to each side of the equation.
3x + -3x + y = 0 + -3x

Combine like terms: 3x + -3x = 0
0 + y = 0 + -3x
y = 0 + -3x
Remove the zero:
y = -3x

Add '-1y' to each side of the equation.
y + -1y = -3x + -1y

Combine like terms: y + -1y = 0
0 = -3x + -1y

Simplifying
0 = -3x + -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 3
Set the factor '(x + 2y)' equal to zero and attempt to solve:

Simplifying
x + 2y = 0

Solving
x + 2y = 0

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

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

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

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

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

Simplifying
0 = -1x + -2y

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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