Algebra Examples

$\frac{4 y}{3 x^{2} + 2 x y} - \frac{9 x}{3 x y + 2 y^{2}}$

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $3 x^{2} + 2 x y$ .

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Step 1.1.1

Factor $x$ out of $3 x^{2}$ .

$\frac{4 y}{x (3 x) + 2 x y} - \frac{9 x}{3 x y + 2 y^{2}}$

Step 1.1.2

Factor $x$ out of $2 x y$ .

$\frac{4 y}{x (3 x) + x (2 y)} - \frac{9 x}{3 x y + 2 y^{2}}$

Step 1.1.3

Factor $x$ out of $x (3 x) + x (2 y)$ .

$\frac{4 y}{x (3 x + 2 y)} - \frac{9 x}{3 x y + 2 y^{2}}$

Step 1.2

Factor $y$ out of $3 x y + 2 y^{2}$ .

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Step 1.2.1

Factor $y$ out of $3 x y$ .

$\frac{4 y}{x (3 x + 2 y)} - \frac{9 x}{y (3 x) + 2 y^{2}}$

Step 1.2.2

Factor $y$ out of $2 y^{2}$ .

$\frac{4 y}{x (3 x + 2 y)} - \frac{9 x}{y (3 x) + y (2 y)}$

Step 1.2.3

Factor $y$ out of $y (3 x) + y (2 y)$ .

$\frac{4 y}{x (3 x + 2 y)} - \frac{9 x}{y (3 x + 2 y)}$

Step 2

To write $\frac{4 y}{x (3 x + 2 y)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{4 y}{x (3 x + 2 y)} \cdot \frac{y}{y} - \frac{9 x}{y (3 x + 2 y)}$

Step 3

To write $- \frac{9 x}{y (3 x + 2 y)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 y}{x (3 x + 2 y)} \cdot \frac{y}{y} - \frac{9 x}{y (3 x + 2 y)} \cdot \frac{x}{x}$

Step 4

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

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Step 4.1

Multiply $\frac{4 y}{x (3 x + 2 y)}$ by $\frac{y}{y}$ .

$\frac{4 y \cdot y}{x (3 x + 2 y) y} - \frac{9 x}{y (3 x + 2 y)} \cdot \frac{x}{x}$

Step 4.2

Multiply $\frac{9 x}{y (3 x + 2 y)}$ by $\frac{x}{x}$ .

$\frac{4 y \cdot y}{x (3 x + 2 y) y} - \frac{9 x \cdot x}{y (3 x + 2 y) x}$

Step 4.3

Reorder the factors of $x (3 x + 2 y) y$ .

$\frac{4 y \cdot y}{x y (3 x + 2 y)} - \frac{9 x \cdot x}{y (3 x + 2 y) x}$

Step 4.4

Reorder the factors of $y (3 x + 2 y) x$ .

$\frac{4 y \cdot y}{x y (3 x + 2 y)} - \frac{9 x \cdot x}{x y (3 x + 2 y)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 y \cdot y - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{4 (y^{1} y) - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6.2

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

$\frac{4 (y^{1} y^{1}) - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6.3

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

$\frac{4 y^{1 + 1} - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6.4

Add $1$ and $1$ .

$\frac{4 y^{2} - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6.5

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

$\frac{{(2 y)}^{2} - 9 x \cdot x}{x y (3 x + 2 y)}$

Step 6.6

Rewrite $9 x \cdot x$ as ${(3 x)}^{2}$ .

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

Step 6.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 y$ and $b = 3 x$ .

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

Step 6.8

Multiply $3$ by $- 1$ .

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

Step 7

Cancel the common factor of $2 y + 3 x$ and $3 x + 2 y$ .

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Step 7.1

Reorder terms.

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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