Solve for y a(b-y)-a(b-1)=a(ay-b)

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

Step 1

Simplify $a (b - y) - a (b - 1)$ .

Simplify each term.

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

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

Apply the distributive property.

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

Multiply $- a \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $a$ by $1$ .

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

Combine the opposite terms in $a b - a y - a b + a$ .

Subtract $a b$ from $a b$ .

$- a y + 0 + a = a (a y - b)$

Add $- a y$ and $0$ .

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

Step 2

Simplify $a (a y - b)$ .

Apply the distributive property.

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

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

$- a y + a = a^{2} y + a (- b)$

Rewrite using the commutative property of multiplication.

$- a y + a = a^{2} y - a b$

Step 3

Subtract $a^{2} y$ from both sides of the equation.

$- a y + a - a^{2} y = - a b$

Step 4

Subtract $a$ from both sides of the equation.

$- a y - a^{2} y = - a b - a$

Step 5

Factor $a y$ out of $- a y - a^{2} y$ .

Factor $a y$ out of $- a y$ .

$a y (- 1) - a^{2} y = - a b - a$

Factor $a y$ out of $- a^{2} y$ .

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

Factor $a y$ out of $a y (- 1) + a y (- a)$ .

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

Step 6

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

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

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

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 1 - a$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $a$ .

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Cancel the common factor of $a$ .

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Simplify terms.

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

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

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

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

Factor $- 1$ out of $- a$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Solve for y a(b-y)-a(b-1)=a(ay-b)

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