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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

Reorder the factors of $(y - x) \cdot 2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine the opposite terms in $x y + x (- x) + y \cdot y + y (- x)$ .

Reorder the factors in the terms $x y$ and $y (- x)$ .

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

Subtract $x y$ from $x y$ .

$\frac{x (- x) + y \cdot y + 0 - (5 - x^{2}) \cdot 2}{2 (y - x)}$

Add $x (- x) + y \cdot y$ and $0$ .

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $y$ by $y$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $5$ .

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

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{2}$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 5$ by $2$ .

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

Move $2$ to the left of $x^{2}$ .

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

Add $- x^{2}$ and $2 x^{2}$ .

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

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

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