Precalculus Examples

$\frac{8 a - 3 b}{7 a} - \frac{2 a + 5 b}{4 b}$

Step 1

To write $\frac{8 a - 3 b}{7 a}$ as a fraction with a common denominator, multiply by $\frac{4 b}{4 b}$ .

$\frac{8 a - 3 b}{7 a} \cdot \frac{4 b}{4 b} - \frac{2 a + 5 b}{4 b}$

Step 2

To write $- \frac{2 a + 5 b}{4 b}$ as a fraction with a common denominator, multiply by $\frac{7 a}{7 a}$ .

$\frac{8 a - 3 b}{7 a} \cdot \frac{4 b}{4 b} - \frac{2 a + 5 b}{4 b} \cdot \frac{7 a}{7 a}$

Step 3

Write each expression with a common denominator of $28 a b$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8 a - 3 b}{7 a}$ by $\frac{4 b}{4 b}$ .

$\frac{(8 a - 3 b) (4 b)}{7 a (4 b)} - \frac{2 a + 5 b}{4 b} \cdot \frac{7 a}{7 a}$

Step 3.2

Multiply $4$ by $7$ .

$\frac{(8 a - 3 b) (4 b)}{28 a b} - \frac{2 a + 5 b}{4 b} \cdot \frac{7 a}{7 a}$

Step 3.3

Multiply $\frac{2 a + 5 b}{4 b}$ by $\frac{7 a}{7 a}$ .

$\frac{(8 a - 3 b) (4 b)}{28 a b} - \frac{(2 a + 5 b) (7 a)}{4 b (7 a)}$

Step 3.4

Multiply $7$ by $4$ .

$\frac{(8 a - 3 b) (4 b)}{28 a b} - \frac{(2 a + 5 b) (7 a)}{28 b a}$

Step 3.5

Reorder the factors of $28 b a$ .

$\frac{(8 a - 3 b) (4 b)}{28 a b} - \frac{(2 a + 5 b) (7 a)}{28 a b}$

Step 4

Combine the numerators over the common denominator.

$\frac{(8 a - 3 b) (4 b) - (2 a + 5 b) (7 a)}{28 a b}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(8 a \cdot 4 - 3 b \cdot 4) b - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.2

Multiply $4$ by $8$ .

$\frac{(32 a - 3 b \cdot 4) b - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.3

Multiply $4$ by $- 3$ .

$\frac{(32 a - 12 b) b - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.4

Apply the distributive property.

$\frac{32 a b - 12 b \cdot b - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.5

Multiply $b$ by $b$ by adding the exponents.

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Step 5.5.1

Move $b$ .

$\frac{32 a b - 12 (b \cdot b) - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.5.2

Multiply $b$ by $b$ .

$\frac{32 a b - 12 b^{2} - (2 a + 5 b) (7 a)}{28 a b}$

Step 5.6

Apply the distributive property.

$\frac{32 a b - 12 b^{2} + (- (2 a) - (5 b)) (7 a)}{28 a b}$

Step 5.7

Multiply $2$ by $- 1$ .

$\frac{32 a b - 12 b^{2} + (- 2 a - (5 b)) (7 a)}{28 a b}$

Step 5.8

Multiply $5$ by $- 1$ .

$\frac{32 a b - 12 b^{2} + (- 2 a - 5 b) (7 a)}{28 a b}$

Step 5.9

Apply the distributive property.

$\frac{32 a b - 12 b^{2} - 2 a (7 a) - 5 b (7 a)}{28 a b}$

Step 5.10

Rewrite using the commutative property of multiplication.

$\frac{32 a b - 12 b^{2} - 2 \cdot 7 a \cdot a - 5 b (7 a)}{28 a b}$

Step 5.11

Rewrite using the commutative property of multiplication.

$\frac{32 a b - 12 b^{2} - 2 \cdot 7 a \cdot a - 5 \cdot 7 b a}{28 a b}$

Step 5.12

Simplify each term.

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Step 5.12.1

Multiply $a$ by $a$ by adding the exponents.

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Step 5.12.1.1

Move $a$ .

$\frac{32 a b - 12 b^{2} - 2 \cdot 7 (a \cdot a) - 5 \cdot 7 b a}{28 a b}$

Step 5.12.1.2

Multiply $a$ by $a$ .

$\frac{32 a b - 12 b^{2} - 2 \cdot 7 a^{2} - 5 \cdot 7 b a}{28 a b}$

Step 5.12.2

Multiply $- 2$ by $7$ .

$\frac{32 a b - 12 b^{2} - 14 a^{2} - 5 \cdot 7 b a}{28 a b}$

Step 5.12.3

Multiply $- 5$ by $7$ .

$\frac{32 a b - 12 b^{2} - 14 a^{2} - 35 b a}{28 a b}$

Step 5.13

Subtract $35 b a$ from $32 a b$ .

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Step 5.13.1

Move $b$ .

$\frac{- 12 b^{2} - 14 a^{2} + 32 a b - 35 a b}{28 a b}$

Step 5.13.2

Subtract $35 a b$ from $32 a b$ .

$\frac{- 12 b^{2} - 14 a^{2} - 3 a b}{28 a b}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 12 b^{2}$ .

$\frac{- (12 b^{2}) - 14 a^{2} - 3 a b}{28 a b}$

Step 6.2

Factor $- 1$ out of $- 14 a^{2}$ .

$\frac{- (12 b^{2}) - (14 a^{2}) - 3 a b}{28 a b}$

Step 6.3

Factor $- 1$ out of $- (12 b^{2}) - (14 a^{2})$ .

$\frac{- (12 b^{2} + 14 a^{2}) - 3 a b}{28 a b}$

Step 6.4

Factor $- 1$ out of $- 3 a b$ .

$\frac{- (12 b^{2} + 14 a^{2}) - (3 a b)}{28 a b}$

Step 6.5

Factor $- 1$ out of $- (12 b^{2} + 14 a^{2}) - (3 a b)$ .

$\frac{- (12 b^{2} + 14 a^{2} + 3 a b)}{28 a b}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (12 b^{2} + 14 a^{2} + 3 a b)$ as $- 1 (12 b^{2} + 14 a^{2} + 3 a b)$ .

$\frac{- 1 (12 b^{2} + 14 a^{2} + 3 a b)}{28 a b}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{12 b^{2} + 14 a^{2} + 3 a b}{28 a b}$