a(x-y)-a(b-1)=a(ay-b)

Solution for a(x-y)-a(b-1)=a(ay-b) equation:

Simplifying
a(x + -1y) + -1a(b + -1) = a(ay + -1b)
(x * a + -1y * a) + -1a(b + -1) = a(ay + -1b)
(ax + -1ay) + -1a(b + -1) = a(ay + -1b)

Reorder the terms:
ax + -1ay + -1a(-1 + b) = a(ay + -1b)
ax + -1ay + (-1 * -1a + b * -1a) = a(ay + -1b)
ax + -1ay + (1a + -1ab) = a(ay + -1b)

Reorder the terms:
1a + -1ab + ax + -1ay = a(ay + -1b)
1a + -1ab + ax + -1ay = (ay * a + -1b * a)

Reorder the terms:
1a + -1ab + ax + -1ay = (-1ab + a2y)
1a + -1ab + ax + -1ay = (-1ab + a2y)

Add 'ab' to each side of the equation.
1a + -1ab + ax + ab + -1ay = -1ab + ab + a2y

Reorder the terms:
1a + -1ab + ab + ax + -1ay = -1ab + ab + a2y

Combine like terms: -1ab + ab = 0
1a + 0 + ax + -1ay = -1ab + ab + a2y
1a + ax + -1ay = -1ab + ab + a2y

Combine like terms: -1ab + ab = 0
1a + ax + -1ay = 0 + a2y
1a + ax + -1ay = a2y

Solving
1a + ax + -1ay = a2y

Solving for variable 'a'.

Combine like terms: a2y + -1a2y = 0
1a + ax + -1ay + -1a2y = 0

Factor out the Greatest Common Factor (GCF), 'a'.
a(1 + x + -1y + -1ay) = 0

Subproblem 1
Set the factor 'a' equal to zero and attempt to solve:

Simplifying
a = 0

Solving
a = 0

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

Simplifying
a = 0
Subproblem 2
Set the factor '(1 + x + -1y + -1ay)' equal to zero and attempt to solve:

Simplifying
1 + x + -1y + -1ay = 0

Reorder the terms:
1 + -1ay + x + -1y = 0

Solving
1 + -1ay + x + -1y = 0

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

Add '-1' to each side of the equation.
1 + -1ay + x + -1 + -1y = 0 + -1

Reorder the terms:
1 + -1 + -1ay + x + -1y = 0 + -1

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

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

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

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

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

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

Divide each side by '-1y'.
a = y-1 + xy-1 + -1

Simplifying
a = y-1 + xy-1 + -1

Reorder the terms:
a = -1 + xy-1 + y-1
Solution
a = {0, -1 + xy-1 + y-1}