a[b-x]-a[b-1]=a[ax-p]

Solution for a[b-x]-a[b-1]=a[ax-p] equation:

Simplifying
a[b + -1x] + -1a[b + -1] = a[ax + -1p]
[b * a + -1x * a] + -1a[b + -1] = a[ax + -1p]
[ab + -1ax] + -1a[b + -1] = a[ax + -1p]

Reorder the terms:
ab + -1ax + -1a[-1 + b] = a[ax + -1p]
ab + -1ax + [-1 * -1a + b * -1a] = a[ax + -1p]
ab + -1ax + [1a + -1ab] = a[ax + -1p]

Reorder the terms:
1a + ab + -1ab + -1ax = a[ax + -1p]

Combine like terms: ab + -1ab = 0
1a + 0 + -1ax = a[ax + -1p]
1a + -1ax = a[ax + -1p]
1a + -1ax = [ax * a + -1p * a]

Reorder the terms:
1a + -1ax = [-1ap + a2x]
1a + -1ax = [-1ap + a2x]

Solving
1a + -1ax = -1ap + a2x

Solving for variable 'a'.

Reorder the terms:
1a + ap + -1ax + -1a2x = -1ap + a2x + ap + -1a2x

Reorder the terms:
1a + ap + -1ax + -1a2x = -1ap + ap + a2x + -1a2x

Combine like terms: -1ap + ap = 0
1a + ap + -1ax + -1a2x = 0 + a2x + -1a2x
1a + ap + -1ax + -1a2x = a2x + -1a2x

Combine like terms: a2x + -1a2x = 0
1a + ap + -1ax + -1a2x = 0

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

Simplifying
1 + p + -1x + -1ax = 0

Reorder the terms:
1 + -1ax + p + -1x = 0

Solving
1 + -1ax + p + -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 + -1ax + p + -1 + -1x = 0 + -1

Reorder the terms:
1 + -1 + -1ax + p + -1x = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -1ax + p + -1x = 0 + -1
-1ax + p + -1x = 0 + -1

Combine like terms: 0 + -1 = -1
-1ax + p + -1x = -1

Add '-1p' to each side of the equation.
-1ax + p + -1p + -1x = -1 + -1p

Combine like terms: p + -1p = 0
-1ax + 0 + -1x = -1 + -1p
-1ax + -1x = -1 + -1p

Add 'x' to each side of the equation.
-1ax + -1x + x = -1 + -1p + x

Combine like terms: -1x + x = 0
-1ax + 0 = -1 + -1p + x
-1ax = -1 + -1p + x

Divide each side by '-1x'.
a = x-1 + px-1 + -1

Simplifying
a = x-1 + px-1 + -1

Reorder the terms:
a = -1 + px-1 + x-1
Solution
a = {0, -1 + px-1 + x-1}