Simplify ((4x^2-y^2)/(xy))/(2/y-1/x)

$\frac{\frac{4 x^{2} - y^{2}}{x y}}{\frac{2}{y} - \frac{1}{x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 x^{2} - y^{2}}{x y} \cdot \frac{1}{\frac{2}{y} - \frac{1}{x}}$

Step 2

Simplify the numerator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

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

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

Step 3

Simplify the denominator.

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

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

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

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

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

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

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

Multiply $\frac{1}{x}$ by $\frac{y}{y}$ .

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{x y}{2 x - y}$ by $1$ .

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

Step 6

Cancel the common factor of $2 x - y$ .

Factor $2 x - y$ out of $(2 x + y) (2 x - y)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x + y}{x y} (x y)$

Step 7

Cancel the common factor of $x y$ .

Cancel the common factor.

$\frac{2 x + y}{x y} (x y)$

Rewrite the expression.

$2 x + y$