(ax+ky+c)dx+(kx+by+c)dy=0

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Solution for (ax+ky+c)dx+(kx+by+c)dy=0 equation:


Simplifying
(ax + ky + c) * dx + (kx + by + c) * dy = 0

Reorder the terms:
(ax + c + ky) * dx + (kx + by + c) * dy = 0

Reorder the terms for easier multiplication:
dx(ax + c + ky) + (kx + by + c) * dy = 0
(ax * dx + c * dx + ky * dx) + (kx + by + c) * dy = 0
(adx2 + cdx + dkxy) + (kx + by + c) * dy = 0

Reorder the terms:
adx2 + cdx + dkxy + (by + c + kx) * dy = 0

Reorder the terms for easier multiplication:
adx2 + cdx + dkxy + dy(by + c + kx) = 0
adx2 + cdx + dkxy + (by * dy + c * dy + kx * dy) = 0
adx2 + cdx + dkxy + (bdy2 + cdy + dkxy) = 0

Reorder the terms:
adx2 + bdy2 + cdx + cdy + dkxy + dkxy = 0

Combine like terms: dkxy + dkxy = 2dkxy
adx2 + bdy2 + cdx + cdy + 2dkxy = 0

Solving
adx2 + bdy2 + cdx + cdy + 2dkxy = 0

Solving for variable 'a'.

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

Add '-1bdy2' to each side of the equation.
adx2 + bdy2 + cdx + cdy + -1bdy2 + 2dkxy = 0 + -1bdy2

Reorder the terms:
adx2 + bdy2 + -1bdy2 + cdx + cdy + 2dkxy = 0 + -1bdy2

Combine like terms: bdy2 + -1bdy2 = 0
adx2 + 0 + cdx + cdy + 2dkxy = 0 + -1bdy2
adx2 + cdx + cdy + 2dkxy = 0 + -1bdy2
Remove the zero:
adx2 + cdx + cdy + 2dkxy = -1bdy2

Add '-1cdx' to each side of the equation.
adx2 + cdx + cdy + -1cdx + 2dkxy = -1bdy2 + -1cdx

Reorder the terms:
adx2 + cdx + -1cdx + cdy + 2dkxy = -1bdy2 + -1cdx

Combine like terms: cdx + -1cdx = 0
adx2 + 0 + cdy + 2dkxy = -1bdy2 + -1cdx
adx2 + cdy + 2dkxy = -1bdy2 + -1cdx

Add '-1cdy' to each side of the equation.
adx2 + cdy + -1cdy + 2dkxy = -1bdy2 + -1cdx + -1cdy

Combine like terms: cdy + -1cdy = 0
adx2 + 0 + 2dkxy = -1bdy2 + -1cdx + -1cdy
adx2 + 2dkxy = -1bdy2 + -1cdx + -1cdy

Add '-2dkxy' to each side of the equation.
adx2 + 2dkxy + -2dkxy = -1bdy2 + -1cdx + -1cdy + -2dkxy

Combine like terms: 2dkxy + -2dkxy = 0
adx2 + 0 = -1bdy2 + -1cdx + -1cdy + -2dkxy
adx2 = -1bdy2 + -1cdx + -1cdy + -2dkxy

Divide each side by 'dx2'.
a = -1bx-2y2 + -1cx-1 + -1cx-2y + -2kx-1y

Simplifying
a = -1bx-2y2 + -1cx-1 + -1cx-2y + -2kx-1y

Reorder the terms:
a = -1bx-2y2 + -1cx-2y + -1cx-1 + -2kx-1y

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