-(4x+3y)dx=(2x+y)dy

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Solution for -(4x+3y)dx=(2x+y)dy equation: 


Simplifying
-1(4x + 3y) * dx = (2x + y) * dy

Reorder the terms for easier multiplication:
-1dx(4x + 3y) = (2x + y) * dy
(4x * -1dx + 3y * -1dx) = (2x + y) * dy

Reorder the terms:
(-3dxy + -4dx2) = (2x + y) * dy
(-3dxy + -4dx2) = (2x + y) * dy

Reorder the terms for easier multiplication:
-3dxy + -4dx2 = dy(2x + y)
-3dxy + -4dx2 = (2x * dy + y * dy)
-3dxy + -4dx2 = (2dxy + dy2)

Solving
-3dxy + -4dx2 = 2dxy + dy2

Solving for variable 'd'.

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

Add '-2dxy' to each side of the equation.
-3dxy + -2dxy + -4dx2 = 2dxy + -2dxy + dy2

Combine like terms: -3dxy + -2dxy = -5dxy
-5dxy + -4dx2 = 2dxy + -2dxy + dy2

Combine like terms: 2dxy + -2dxy = 0
-5dxy + -4dx2 = 0 + dy2
-5dxy + -4dx2 = dy2

Add '-1dy2' to each side of the equation.
-5dxy + -4dx2 + -1dy2 = dy2 + -1dy2

Combine like terms: dy2 + -1dy2 = 0
-5dxy + -4dx2 + -1dy2 = 0

Factor out the Greatest Common Factor (GCF), '-1d'.
-1d(5xy + 4x2 + y2) = 0

Factor a trinomial.
-1d((4x + y)(x + y)) = 0

Ignore the factor -1.

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 '(4x + y)' equal to zero and attempt to solve:

Simplifying
4x + y = 0

Solving
4x + y = 0

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

Add '-4x' to each side of the equation.
4x + -4x + y = 0 + -4x

Combine like terms: 4x + -4x = 0
0 + y = 0 + -4x
y = 0 + -4x
Remove the zero:
y = -4x

Add '-1y' to each side of the equation.
y + -1y = -4x + -1y

Combine like terms: y + -1y = 0
0 = -4x + -1y

Simplifying
0 = -4x + -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 + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

Solving
x + y = 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 + y = 0 + -1x

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

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

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

Simplifying
0 = -1x + -1y

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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