Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $x^{2} y^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $x$ by $x$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $2$ by $- 1$ .

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

Step 5.5

Multiply $- - x$ .

Tap for more steps...

Step 5.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.5.2

Multiply $x$ by $1$ .

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

Step 5.6

Apply the distributive property.

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

Step 5.7

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 5.7.1

Move $y$ .

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

Step 5.7.2

Multiply $y$ by $y$ .

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

Step 5.8

Add $- 2 y x$ and $x y$ .

Tap for more steps...

Step 5.8.1

Move $y$ .

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

Step 5.8.2

Add $- 2 x y$ and $x y$ .

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

Step 5.9

Factor by grouping.

Tap for more steps...

Step 5.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot - 2 = - 2$ and whose sum is $b = - 1$ .

Tap for more steps...

Step 5.9.1.1

Reorder $- 2 y^{2}$ and $- x y$ .

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

Step 5.9.1.2

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

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

Step 5.9.1.3

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 5.9.1.4

Apply the distributive property.

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

Step 5.9.1.5

Multiply $x y$ by $1$ .

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

Step 5.9.1.6

Move parentheses.

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

Step 5.9.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.9.2.1

Group the first two terms and the last two terms.

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

Step 5.9.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.9.3

Factor the polynomial by factoring out the greatest common factor, $x + y$ .

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