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

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

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

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

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

Combine like terms: ab + -1ab = 0
1a + 0 + -1ax = a(ay + -1b)
1a + -1ax = a(ay + -1b)
1a + -1ax = (ay * a + -1b * a)

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

Solving
1a + -1ax = -1ab + a2y

Solving for variable 'a'.

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

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

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

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

Factor out the Greatest Common Factor (GCF), 'a'.
a(1 + b + -1x + -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 + b + -1x + -1ay)' equal to zero and attempt to solve:

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

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

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

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

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

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

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

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

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

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

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

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

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