Solve for a ab-a=-b+1

$a b - a = - b + 1$

Factor $a$ out of $a b - a$ .

Factor $a$ out of $a b$ .

$a (b) - a = - b + 1$

Factor $a$ out of $- a$ .

$a (b) + a \cdot - 1 = - b + 1$

Factor $a$ out of $a (b) + a \cdot - 1$ .

$a (b - 1) = - b + 1$

Divide each term in $a (b - 1) = - b + 1$ by $b - 1$ and simplify.

Divide each term in $a (b - 1) = - b + 1$ by $b - 1$ .

$\frac{a (b - 1)}{b - 1} = \frac{- b}{b - 1} + \frac{1}{b - 1}$

Simplify the left side.

Cancel the common factor of $b - 1$ .

Cancel the common factor.

$\frac{a (b - 1)}{b - 1} = \frac{- b}{b - 1} + \frac{1}{b - 1}$

Divide $a$ by $1$ .

$a = \frac{- b}{b - 1} + \frac{1}{b - 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$a = \frac{- b + 1}{b - 1}$

Cancel the common factor of $- b + 1$ and $b - 1$ .

Factor $- 1$ out of $- b$ .

$a = \frac{- (b) + 1}{b - 1}$

Rewrite $1$ as $- 1 (- 1)$ .

$a = \frac{- (b) - 1 (- 1)}{b - 1}$

Factor $- 1$ out of $- (b) - 1 (- 1)$ .

$a = \frac{- (b - 1)}{b - 1}$

Rewrite $- (b - 1)$ as $- 1 (b - 1)$ .

$a = \frac{- 1 (b - 1)}{b - 1}$

Cancel the common factor.

$a = \frac{- 1 (b - 1)}{b - 1}$

Divide $- 1$ by $1$ .

$a = - 1$