Simplify 3/(ax-ay)+2/(by-bx)

$\frac{3}{a x - a y} + \frac{2}{b y - b x}$

Step 1

Simplify each term.

Simplify the denominator.

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

Factor $a$ out of $a x$ .

$\frac{3}{a (x) - a y} + \frac{2}{b y - b x}$

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

$\frac{3}{a (x) + a (- 1 y)} + \frac{2}{b y - b x}$

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

$\frac{3}{a (x - 1 y)} + \frac{2}{b y - b x}$

Rewrite $- 1 y$ as $- y$ .

$\frac{3}{a (x - y)} + \frac{2}{b y - b x}$

Simplify the denominator.

Factor $b$ out of $b y - b x$ .

Factor $b$ out of $b y$ .

$\frac{3}{a (x - y)} + \frac{2}{b (y) - b x}$

Factor $b$ out of $- b x$ .

$\frac{3}{a (x - y)} + \frac{2}{b (y) + b (- 1 x)}$

Factor $b$ out of $b (y) + b (- 1 x)$ .

$\frac{3}{a (x - y)} + \frac{2}{b (y - 1 x)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{3}{a (x - y)} + \frac{2}{b (y - x)}$

Step 2

Simplify with factoring out.

Factor $- 1$ out of $y$ .

$\frac{3}{a (x - y)} + \frac{2}{b (- 1 (- y) - x)}$

Factor $- 1$ out of $- x$ .

$\frac{3}{a (x - y)} + \frac{2}{b (- 1 (- y) - (x))}$

Factor $- 1$ out of $- 1 (- y) - (x)$ .

$\frac{3}{a (x - y)} + \frac{2}{b (- 1 (- y + x))}$

Simplify the expression.

Move a negative from the denominator of $\frac{2}{b (- 1 (- y + x))}$ to the numerator.

$\frac{3}{a (x - y)} + \frac{- 1 \cdot 2}{b (- y + x)}$

Reorder terms.

$\frac{3}{a (x - y)} + \frac{- 1 \cdot 2}{b (x - y)}$

Step 3

To write $\frac{3}{a (x - y)}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{3}{a (x - y)} \cdot \frac{b}{b} + \frac{- 1 \cdot 2}{b (x - y)}$

Step 4

To write $\frac{- 1 \cdot 2}{b (x - y)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{3}{a (x - y)} \cdot \frac{b}{b} + \frac{- 1 \cdot 2}{b (x - y)} \cdot \frac{a}{a}$

Step 5

Write each expression with a common denominator of $a (x - y) b$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{a (x - y)}$ by $\frac{b}{b}$ .

$\frac{3 b}{a (x - y) b} + \frac{- 1 \cdot 2}{b (x - y)} \cdot \frac{a}{a}$

Multiply $\frac{- 1 \cdot 2}{b (x - y)}$ by $\frac{a}{a}$ .

$\frac{3 b}{a (x - y) b} + \frac{- 1 \cdot 2 a}{b (x - y) a}$

Reorder the factors of $a (x - y) b$ .

$\frac{3 b}{a b (x - y)} + \frac{- 1 \cdot 2 a}{b (x - y) a}$

Reorder the factors of $b (x - y) a$ .

$\frac{3 b}{a b (x - y)} + \frac{- 1 \cdot 2 a}{a b (x - y)}$

Step 6

Combine the numerators over the common denominator.

$\frac{3 b - 1 \cdot 2 a}{a b (x - y)}$

Step 7

Multiply $- 1$ by $2$ .

$\frac{3 b - 2 a}{a b (x - y)}$

Simplify 3/(ax-ay)+2/(by-bx)

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