2(4y+5t)dt+(5y+7t)dy=0

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Solution for 2(4y+5t)dt+(5y+7t)dy=0 equation: 


Simplifying
2(4y + 5t) * dt + (5y + 7t) * dy = 0

Reorder the terms:
2(5t + 4y) * dt + (5y + 7t) * dy = 0

Reorder the terms for easier multiplication:
2dt(5t + 4y) + (5y + 7t) * dy = 0
(5t * 2dt + 4y * 2dt) + (5y + 7t) * dy = 0

Reorder the terms:
(8dty + 10dt2) + (5y + 7t) * dy = 0
(8dty + 10dt2) + (5y + 7t) * dy = 0

Reorder the terms:
8dty + 10dt2 + (7t + 5y) * dy = 0

Reorder the terms for easier multiplication:
8dty + 10dt2 + dy(7t + 5y) = 0
8dty + 10dt2 + (7t * dy + 5y * dy) = 0
8dty + 10dt2 + (7dty + 5dy2) = 0

Reorder the terms:
8dty + 7dty + 10dt2 + 5dy2 = 0

Combine like terms: 8dty + 7dty = 15dty
15dty + 10dt2 + 5dy2 = 0

Solving
15dty + 10dt2 + 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(3ty + 2t2 + y2) = 0

Factor a trinomial.
5d((2t + y)(t + 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 '(2t + y)' equal to zero and attempt to solve:

Simplifying
2t + y = 0

Solving
2t + y = 0

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

Add '-2t' to each side of the equation.
2t + -2t + y = 0 + -2t

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

Add '-1y' to each side of the equation.
y + -1y = -2t + -1y

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

Simplifying
0 = -2t + -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 '(t + y)' equal to zero and attempt to solve:

Simplifying
t + y = 0

Solving
t + y = 0

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

Add '-1t' to each side of the equation.
t + -1t + y = 0 + -1t

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

Add '-1y' to each side of the equation.
y + -1y = -1t + -1y

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

Simplifying
0 = -1t + -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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