a(x+c)+b(x+d)-ax=e

Solution for a(x+c)+b(x+d)-ax=e equation:

Simplifying
a(x + c) + b(x + d) + -1ax = e

Reorder the terms:
a(c + x) + b(x + d) + -1ax = e
(c * a + x * a) + b(x + d) + -1ax = e
(ac + ax) + b(x + d) + -1ax = e

Reorder the terms:
ac + ax + b(d + x) + -1ax = e
ac + ax + (d * b + x * b) + -1ax = e
ac + ax + (bd + bx) + -1ax = e

Reorder the terms:
ac + ax + -1ax + bd + bx = e

Combine like terms: ax + -1ax = 0
ac + 0 + bd + bx = e
ac + bd + bx = e

Solving
ac + bd + bx = e

Solving for variable 'a'.

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

Add '-1bd' to each side of the equation.
ac + bd + -1bd + bx = -1bd + e

Combine like terms: bd + -1bd = 0
ac + 0 + bx = -1bd + e
ac + bx = -1bd + e

Add '-1bx' to each side of the equation.
ac + bx + -1bx = -1bd + -1bx + e

Combine like terms: bx + -1bx = 0
ac + 0 = -1bd + -1bx + e
ac = -1bd + -1bx + e

Divide each side by 'c'.
a = -1bc-1d + -1bc-1x + c-1e

Simplifying
a = -1bc-1d + -1bc-1x + c-1e