(10x+8y)dx+(7x+5y)dy=0

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Solution for (10x+8y)dx+(7x+5y)dy=0 equation: 


Simplifying
(10x + 8y) * dx + (7x + 5y) * dy = 0

Reorder the terms for easier multiplication:
dx(10x + 8y) + (7x + 5y) * dy = 0
(10x * dx + 8y * dx) + (7x + 5y) * dy = 0

Reorder the terms:
(8dxy + 10dx2) + (7x + 5y) * dy = 0
(8dxy + 10dx2) + (7x + 5y) * dy = 0

Reorder the terms for easier multiplication:
8dxy + 10dx2 + dy(7x + 5y) = 0
8dxy + 10dx2 + (7x * dy + 5y * dy) = 0
8dxy + 10dx2 + (7dxy + 5dy2) = 0

Reorder the terms:
8dxy + 7dxy + 10dx2 + 5dy2 = 0

Combine like terms: 8dxy + 7dxy = 15dxy
15dxy + 10dx2 + 5dy2 = 0

Solving
15dxy + 10dx2 + 5dy2 = 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), '5d'.
5d(3xy + 2x2 + y2) = 0

Factor a trinomial.
5d((2x + y)(x + y)) = 0

Ignore the factor 5.

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 '(2x + y)' equal to zero and attempt to solve:

Simplifying
2x + y = 0

Solving
2x + y = 0

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

Add '-2x' to each side of the equation.
2x + -2x + y = 0 + -2x

Combine like terms: 2x + -2x = 0
0 + y = 0 + -2x
y = 0 + -2x
Remove the zero:
y = -2x

Add '-1y' to each side of the equation.
y + -1y = -2x + -1y

Combine like terms: y + -1y = 0
0 = -2x + -1y

Simplifying
0 = -2x + -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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