Simplify (x-y)/(2x)+(5-y)/(3-x)

$\frac{x - y}{2 x} + \frac{5 - y}{3 - x}$

To write $\frac{x - y}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3 - x}{3 - x}$ .

$\frac{x - y}{2 x} \cdot \frac{3 - x}{3 - x} + \frac{5 - y}{3 - x}$

To write $\frac{5 - y}{3 - x}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{x - y}{2 x} \cdot \frac{3 - x}{3 - x} + \frac{5 - y}{3 - x} \cdot \frac{2 x}{2 x}$

Write each expression with a common denominator of $2 x (3 - x)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - y}{2 x}$ by $\frac{3 - x}{3 - x}$ .

$\frac{(x - y) (3 - x)}{2 x (3 - x)} + \frac{5 - y}{3 - x} \cdot \frac{2 x}{2 x}$

Multiply $\frac{5 - y}{3 - x}$ by $\frac{2 x}{2 x}$ .

$\frac{(x - y) (3 - x)}{2 x (3 - x)} + \frac{(5 - y) (2 x)}{(3 - x) (2 x)}$

Reorder the factors of $(3 - x) (2 x)$ .

$\frac{(x - y) (3 - x)}{2 x (3 - x)} + \frac{(5 - y) (2 x)}{2 x (3 - x)}$

Combine the numerators over the common denominator.

$\frac{(x - y) (3 - x) + (5 - y) (2 x)}{2 x (3 - x)}$

Simplify the numerator.

Expand $(x - y) (3 - x)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (3 - x) - y (3 - x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Apply the distributive property.

$\frac{x \cdot 3 + x (- x) - y (3 - x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Apply the distributive property.

$\frac{x \cdot 3 + x (- x) - y \cdot 3 - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Simplify each term.

Move $3$ to the left of $x$ .

$\frac{3 \cdot x + x (- x) - y \cdot 3 - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Rewrite using the commutative property of multiplication.

$\frac{3 x - x \cdot x - y \cdot 3 - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 x - (x \cdot x) - y \cdot 3 - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Multiply $x$ by $x$ .

$\frac{3 x - x^{2} - y \cdot 3 - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Multiply $3$ by $- 1$ .

$\frac{3 x - x^{2} - 3 y - y (- x) + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Rewrite using the commutative property of multiplication.

$\frac{3 x - x^{2} - 3 y - 1 \cdot - 1 y x + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Multiply $- 1$ by $- 1$ .

$\frac{3 x - x^{2} - 3 y + 1 y x + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Multiply $y$ by $1$ .

$\frac{3 x - x^{2} - 3 y + y x + (5 - y) \cdot 2 x}{2 x (3 - x)}$

Apply the distributive property.

$\frac{3 x - x^{2} - 3 y + y x + (5 \cdot 2 - y \cdot 2) x}{2 x (3 - x)}$

Multiply $5$ by $2$ .

$\frac{3 x - x^{2} - 3 y + y x + (10 - y \cdot 2) x}{2 x (3 - x)}$

Multiply $2$ by $- 1$ .

$\frac{3 x - x^{2} - 3 y + y x + (10 - 2 y) x}{2 x (3 - x)}$

Apply the distributive property.

$\frac{3 x - x^{2} - 3 y + y x + 10 x - 2 y x}{2 x (3 - x)}$

Add $3 x$ and $10 x$ .

$\frac{- x^{2} - 3 y + y x + 13 x - 2 y x}{2 x (3 - x)}$

Subtract $2 y x$ from $y x$ .

$\frac{- x^{2} - 3 y - y x + 13 x}{2 x (3 - x)}$

Simplify with factoring out.

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - 3 y - y x + 13 x}{2 x (3 - x)}$

Factor $- 1$ out of $- 3 y$ .

$\frac{- (x^{2}) - (3 y) - y x + 13 x}{2 x (3 - x)}$

Factor $- 1$ out of $- (x^{2}) - (3 y)$ .

$\frac{- (x^{2} + 3 y) - y x + 13 x}{2 x (3 - x)}$

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

$\frac{- (x^{2} + 3 y) - (y x) + 13 x}{2 x (3 - x)}$

Factor $- 1$ out of $- (x^{2} + 3 y) - (y x)$ .

$\frac{- (x^{2} + 3 y + y x) + 13 x}{2 x (3 - x)}$

Factor $- 1$ out of $13 x$ .

$\frac{- (x^{2} + 3 y + y x) - (- 13 x)}{2 x (3 - x)}$

Factor $- 1$ out of $- (x^{2} + 3 y + y x) - (- 13 x)$ .

$\frac{- (x^{2} + 3 y + y x - 13 x)}{2 x (3 - x)}$

Simplify the expression.

Rewrite $- (x^{2} + 3 y + y x - 13 x)$ as $- 1 (x^{2} + 3 y + y x - 13 x)$ .

$\frac{- 1 (x^{2} + 3 y + y x - 13 x)}{2 x (3 - x)}$

Move the negative in front of the fraction.

$- \frac{x^{2} + 3 y + y x - 13 x}{2 x (3 - x)}$

Simplify (x-y)/(2x)+(5-y)/(3-x)

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