(2x-6y+3)dx-(x+3y+1)dy=0

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Solution for (2x-6y+3)dx-(x+3y+1)dy=0 equation:


Simplifying
(2x + -6y + 3) * dx + -1(x + 3y + 1) * dy = 0

Reorder the terms:
(3 + 2x + -6y) * dx + -1(x + 3y + 1) * dy = 0

Reorder the terms for easier multiplication:
dx(3 + 2x + -6y) + -1(x + 3y + 1) * dy = 0
(3 * dx + 2x * dx + -6y * dx) + -1(x + 3y + 1) * dy = 0

Reorder the terms:
(3dx + -6dxy + 2dx2) + -1(x + 3y + 1) * dy = 0
(3dx + -6dxy + 2dx2) + -1(x + 3y + 1) * dy = 0

Reorder the terms:
3dx + -6dxy + 2dx2 + -1(1 + x + 3y) * dy = 0

Reorder the terms for easier multiplication:
3dx + -6dxy + 2dx2 + -1dy(1 + x + 3y) = 0
3dx + -6dxy + 2dx2 + (1 * -1dy + x * -1dy + 3y * -1dy) = 0

Reorder the terms:
3dx + -6dxy + 2dx2 + (-1dxy + -1dy + -3dy2) = 0
3dx + -6dxy + 2dx2 + (-1dxy + -1dy + -3dy2) = 0

Reorder the terms:
3dx + -6dxy + -1dxy + 2dx2 + -1dy + -3dy2 = 0

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

Solving
3dx + -7dxy + 2dx2 + -1dy + -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(3x + -7xy + 2x2 + -1y + -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 '(3x + -7xy + 2x2 + -1y + -3y2)' equal to zero and attempt to solve: Simplifying 3x + -7xy + 2x2 + -1y + -3y2 = 0 Solving 3x + -7xy + 2x2 + -1y + -3y2 = 0 Move all terms containing d to the left, all other terms to the right. Add '-3x' to each side of the equation. 3x + -7xy + 2x2 + -1y + -3x + -3y2 = 0 + -3x Reorder the terms: 3x + -3x + -7xy + 2x2 + -1y + -3y2 = 0 + -3x Combine like terms: 3x + -3x = 0 0 + -7xy + 2x2 + -1y + -3y2 = 0 + -3x -7xy + 2x2 + -1y + -3y2 = 0 + -3x Remove the zero: -7xy + 2x2 + -1y + -3y2 = -3x Add '7xy' to each side of the equation. -7xy + 2x2 + -1y + 7xy + -3y2 = -3x + 7xy Reorder the terms: -7xy + 7xy + 2x2 + -1y + -3y2 = -3x + 7xy Combine like terms: -7xy + 7xy = 0 0 + 2x2 + -1y + -3y2 = -3x + 7xy 2x2 + -1y + -3y2 = -3x + 7xy Add '-2x2' to each side of the equation. 2x2 + -1y + -2x2 + -3y2 = -3x + 7xy + -2x2 Reorder the terms: 2x2 + -2x2 + -1y + -3y2 = -3x + 7xy + -2x2 Combine like terms: 2x2 + -2x2 = 0 0 + -1y + -3y2 = -3x + 7xy + -2x2 -1y + -3y2 = -3x + 7xy + -2x2 Add 'y' to each side of the equation. -1y + y + -3y2 = -3x + 7xy + -2x2 + y Combine like terms: -1y + y = 0 0 + -3y2 = -3x + 7xy + -2x2 + y -3y2 = -3x + 7xy + -2x2 + y Add '3y2' to each side of the equation. -3y2 + 3y2 = -3x + 7xy + -2x2 + y + 3y2 Combine like terms: -3y2 + 3y2 = 0 0 = -3x + 7xy + -2x2 + y + 3y2 Simplifying 0 = -3x + 7xy + -2x2 + y + 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}

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