(2x-6y+1)dx=(x-3y+6)dy

Solution for (2x-6y+1)dx=(x-3y+6)dy equation:

Simplifying
(2x + -6y + 1) * dx = (x + -3y + 6) * dy

Reorder the terms:
(1 + 2x + -6y) * dx = (x + -3y + 6) * dy

Reorder the terms for easier multiplication:
dx(1 + 2x + -6y) = (x + -3y + 6) * dy
(1 * dx + 2x * dx + -6y * dx) = (x + -3y + 6) * dy

Reorder the terms:
(1dx + -6dxy + 2dx2) = (x + -3y + 6) * dy
(1dx + -6dxy + 2dx2) = (x + -3y + 6) * dy

Reorder the terms:
1dx + -6dxy + 2dx2 = (6 + x + -3y) * dy

Reorder the terms for easier multiplication:
1dx + -6dxy + 2dx2 = dy(6 + x + -3y)
1dx + -6dxy + 2dx2 = (6 * dy + x * dy + -3y * dy)

Reorder the terms:
1dx + -6dxy + 2dx2 = (dxy + 6dy + -3dy2)
1dx + -6dxy + 2dx2 = (dxy + 6dy + -3dy2)

Solving
1dx + -6dxy + 2dx2 = dxy + 6dy + -3dy2

Solving for variable 'd'.

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

Add '-1dxy' to each side of the equation.
1dx + -6dxy + -1dxy + 2dx2 = dxy + 6dy + -1dxy + -3dy2

Combine like terms: -6dxy + -1dxy = -7dxy
1dx + -7dxy + 2dx2 = dxy + 6dy + -1dxy + -3dy2

Reorder the terms:
1dx + -7dxy + 2dx2 = dxy + -1dxy + 6dy + -3dy2

Combine like terms: dxy + -1dxy = 0
1dx + -7dxy + 2dx2 = 0 + 6dy + -3dy2
1dx + -7dxy + 2dx2 = 6dy + -3dy2

Add '-6dy' to each side of the equation.
1dx + -7dxy + 2dx2 + -6dy = 6dy + -6dy + -3dy2

Combine like terms: 6dy + -6dy = 0
1dx + -7dxy + 2dx2 + -6dy = 0 + -3dy2
1dx + -7dxy + 2dx2 + -6dy = -3dy2

Add '3dy2' to each side of the equation.
1dx + -7dxy + 2dx2 + -6dy + 3dy2 = -3dy2 + 3dy2

Combine like terms: -3dy2 + 3dy2 = 0
1dx + -7dxy + 2dx2 + -6dy + 3dy2 = 0

Factor out the Greatest Common Factor (GCF), 'd'.
d(x + -7xy + 2x2 + -6y + 3y2) = 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 '(x + -7xy + 2x2 + -6y + 3y2)' equal to zero and attempt to solve:

Simplifying
x + -7xy + 2x2 + -6y + 3y2 = 0

Solving
x + -7xy + 2x2 + -6y + 3y2 = 0

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

Add '-1x' to each side of the equation.
x + -7xy + 2x2 + -6y + -1x + 3y2 = 0 + -1x

Reorder the terms:
x + -1x + -7xy + 2x2 + -6y + 3y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + -7xy + 2x2 + -6y + 3y2 = 0 + -1x
-7xy + 2x2 + -6y + 3y2 = 0 + -1x
Remove the zero:
-7xy + 2x2 + -6y + 3y2 = -1x

Add '7xy' to each side of the equation.
-7xy + 2x2 + -6y + 7xy + 3y2 = -1x + 7xy

Reorder the terms:
-7xy + 7xy + 2x2 + -6y + 3y2 = -1x + 7xy

Combine like terms: -7xy + 7xy = 0
0 + 2x2 + -6y + 3y2 = -1x + 7xy
2x2 + -6y + 3y2 = -1x + 7xy

Add '-2x2' to each side of the equation.
2x2 + -6y + -2x2 + 3y2 = -1x + 7xy + -2x2

Reorder the terms:
2x2 + -2x2 + -6y + 3y2 = -1x + 7xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + -6y + 3y2 = -1x + 7xy + -2x2
-6y + 3y2 = -1x + 7xy + -2x2

Add '6y' to each side of the equation.
-6y + 6y + 3y2 = -1x + 7xy + -2x2 + 6y

Combine like terms: -6y + 6y = 0
0 + 3y2 = -1x + 7xy + -2x2 + 6y
3y2 = -1x + 7xy + -2x2 + 6y

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -1x + 7xy + -2x2 + 6y + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -1x + 7xy + -2x2 + 6y + -3y2

Simplifying
0 = -1x + 7xy + -2x2 + 6y + -3y2

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}