(3x+y)dx+(6x+2y)dy=0

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Solution for (3x+y)dx+(6x+2y)dy=0 equation: 


Simplifying
(3x + y) * dx + (6x + 2y) * dy = 0

Reorder the terms for easier multiplication:
dx(3x + y) + (6x + 2y) * dy = 0
(3x * dx + y * dx) + (6x + 2y) * dy = 0

Reorder the terms:
(dxy + 3dx2) + (6x + 2y) * dy = 0
(dxy + 3dx2) + (6x + 2y) * dy = 0

Reorder the terms for easier multiplication:
dxy + 3dx2 + dy(6x + 2y) = 0
dxy + 3dx2 + (6x * dy + 2y * dy) = 0
dxy + 3dx2 + (6dxy + 2dy2) = 0

Reorder the terms:
dxy + 6dxy + 3dx2 + 2dy2 = 0

Combine like terms: dxy + 6dxy = 7dxy
7dxy + 3dx2 + 2dy2 = 0

Solving
7dxy + 3dx2 + 2dy2 = 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(7xy + 3x2 + 2y2) = 0

Factor a trinomial.
d((3x + y)(x + 2y)) = 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 + 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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