(5x-y)dy+(3y-7x)dx=0

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Solution for (5x-y)dy+(3y-7x)dx=0 equation: 


Simplifying
(5x + -1y) * dy + (3y + -7x) * dx = 0

Reorder the terms for easier multiplication:
dy(5x + -1y) + (3y + -7x) * dx = 0
(5x * dy + -1y * dy) + (3y + -7x) * dx = 0
(5dxy + -1dy2) + (3y + -7x) * dx = 0

Reorder the terms:
5dxy + -1dy2 + (-7x + 3y) * dx = 0

Reorder the terms for easier multiplication:
5dxy + -1dy2 + dx(-7x + 3y) = 0
5dxy + -1dy2 + (-7x * dx + 3y * dx) = 0

Reorder the terms:
5dxy + -1dy2 + (3dxy + -7dx2) = 0
5dxy + -1dy2 + (3dxy + -7dx2) = 0

Reorder the terms:
5dxy + 3dxy + -7dx2 + -1dy2 = 0

Combine like terms: 5dxy + 3dxy = 8dxy
8dxy + -7dx2 + -1dy2 = 0

Solving
8dxy + -7dx2 + -1dy2 = 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(8xy + -7x2 + -1y2) = 0

Factor a trinomial.
d((-7x + y)(x + -1y)) = 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 '(-7x + y)' equal to zero and attempt to solve:

Simplifying
-7x + y = 0

Solving
-7x + y = 0

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

Add '7x' to each side of the equation.
-7x + 7x + y = 0 + 7x

Combine like terms: -7x + 7x = 0
0 + y = 0 + 7x
y = 0 + 7x
Remove the zero:
y = 7x

Add '-1y' to each side of the equation.
y + -1y = 7x + -1y

Combine like terms: y + -1y = 0
0 = 7x + -1y

Simplifying
0 = 7x + -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 + -1y)' equal to zero and attempt to solve:

Simplifying
x + -1y = 0

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

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

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

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

Simplifying
0 = -1x + 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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