Simplify (y/x-x/y)/(5/y-5/x)

$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{5}{y} - \frac{5}{x}}$

Multiply the numerator and denominator of the complex fraction by $x y$ .

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

$\frac{x y}{x y} \cdot \frac{\frac{y}{x} - \frac{x}{y}}{\frac{5}{y} - \frac{5}{x}}$

Combine.

$\frac{x y (\frac{y}{x} - \frac{x}{y})}{x y (\frac{5}{y} - \frac{5}{x})}$

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Raise $y$ to the power of $1$ .

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

Raise $y$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{x}{y}$ into the numerator.

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

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{5}{x}$ into the numerator.

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

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = x$ .

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

Simplify the denominator.

Factor $5$ out of $x \cdot 5 + y \cdot - 5$ .

Factor $5$ out of $x \cdot 5$ .

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

Factor $5$ out of $y \cdot - 5$ .

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

Factor $5$ out of $5 \cdot x + 5 (y \cdot - 1)$ .

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

Move $- 1$ to the left of $y$ .

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $y - x$ and $x - y$ .

Factor $- 1$ out of $y$ .

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

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Move $- 1$ to the left of $y + x$ .

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

Move the negative in front of the fraction.

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

Simplify (y/x-x/y)/(5/y-5/x)

Related Questions

Support LCMGCF.com by disabling your adblocker.