Precalculus Examples

$\frac{x - 4}{5 x y} - \frac{3 x + 1}{y^{2}}$

Step 1

To write $\frac{x - 4}{5 x y}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{x - 4}{5 x y} \cdot \frac{y}{y} - \frac{3 x + 1}{y^{2}}$

Step 2

To write $- \frac{3 x + 1}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{5 x}{5 x}$ .

$\frac{x - 4}{5 x y} \cdot \frac{y}{y} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3

Write each expression with a common denominator of $5 x y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x - 4}{5 x y}$ by $\frac{y}{y}$ .

$\frac{(x - 4) y}{5 x y \cdot y} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3.2

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

$\frac{(x - 4) y}{5 x (y^{1} y)} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3.3

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

$\frac{(x - 4) y}{5 x (y^{1} y^{1})} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3.4

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

$\frac{(x - 4) y}{5 x y^{1 + 1}} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3.5

Add $1$ and $1$ .

$\frac{(x - 4) y}{5 x y^{2}} - \frac{3 x + 1}{y^{2}} \cdot \frac{5 x}{5 x}$

Step 3.6

Multiply $\frac{3 x + 1}{y^{2}}$ by $\frac{5 x}{5 x}$ .

$\frac{(x - 4) y}{5 x y^{2}} - \frac{(3 x + 1) (5 x)}{y^{2} (5 x)}$

Step 3.7

Reorder the factors of $5 x y^{2}$ .

$\frac{(x - 4) y}{5 y^{2} x} - \frac{(3 x + 1) (5 x)}{y^{2} (5 x)}$

Step 3.8

Reorder the factors of $y^{2} (5 x)$ .

$\frac{(x - 4) y}{5 y^{2} x} - \frac{(3 x + 1) (5 x)}{5 y^{2} x}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x - 4) y - (3 x + 1) (5 x)}{5 y^{2} x}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x y - 4 y - (3 x + 1) (5 x)}{5 y^{2} x}$

Step 5.2

Apply the distributive property.

$\frac{x y - 4 y + (- (3 x) - 1 \cdot 1) (5 x)}{5 y^{2} x}$

Step 5.3

Multiply $3$ by $- 1$ .

$\frac{x y - 4 y + (- 3 x - 1 \cdot 1) (5 x)}{5 y^{2} x}$

Step 5.4

Multiply $- 1$ by $1$ .

$\frac{x y - 4 y + (- 3 x - 1) (5 x)}{5 y^{2} x}$

Step 5.5

Apply the distributive property.

$\frac{x y - 4 y - 3 x (5 x) - 1 (5 x)}{5 y^{2} x}$

Step 5.6

Rewrite using the commutative property of multiplication.

$\frac{x y - 4 y - 3 \cdot 5 x \cdot x - 1 (5 x)}{5 y^{2} x}$

Step 5.7

Multiply $5$ by $- 1$ .

$\frac{x y - 4 y - 3 \cdot 5 x \cdot x - 5 x}{5 y^{2} x}$

Step 5.8

Simplify each term.

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Step 5.8.1

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

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Step 5.8.1.1

Move $x$ .

$\frac{x y - 4 y - 3 \cdot 5 (x \cdot x) - 5 x}{5 y^{2} x}$

Step 5.8.1.2

Multiply $x$ by $x$ .

$\frac{x y - 4 y - 3 \cdot 5 x^{2} - 5 x}{5 y^{2} x}$

Step 5.8.2

Multiply $- 3$ by $5$ .

$\frac{x y - 4 y - 15 x^{2} - 5 x}{5 y^{2} x}$