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

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

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

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

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

Reorder the terms:
(5dxy + -1dx2) + (3y + -7x) * dy = 0
(5dxy + -1dx2) + (3y + -7x) * dy = 0

Reorder the terms:
5dxy + -1dx2 + (-7x + 3y) * dy = 0

Reorder the terms for easier multiplication:
5dxy + -1dx2 + dy(-7x + 3y) = 0
5dxy + -1dx2 + (-7x * dy + 3y * dy) = 0
5dxy + -1dx2 + (-7dxy + 3dy2) = 0

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

Combine like terms: 5dxy + -7dxy = -2dxy
-2dxy + -1dx2 + 3dy2 = 0

Solving
-2dxy + -1dx2 + 3dy2 = 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(-2xy + -1x2 + 3y2) = 0

Factor a trinomial.
d((-1x + -3y)(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 '(-1x + -3y)' equal to zero and attempt to solve:

Simplifying
-1x + -3y = 0

Solving
-1x + -3y = 0

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

Add 'x' to each side of the equation.
-1x + x + -3y = 0 + x

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

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

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

Simplifying
0 = x + 3y

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}