Algebra Examples

$\frac{5}{a} - \frac{3}{b} + \frac{1}{a b}$

Step 1

To write $\frac{5}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{5}{a} \cdot \frac{b}{b} - \frac{3}{b} + \frac{1}{a b}$

Step 2

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

$\frac{5}{a} \cdot \frac{b}{b} - \frac{3}{b} \cdot \frac{a}{a} + \frac{1}{a b}$

Step 3

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

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Step 3.1

Multiply $\frac{5}{a}$ by $\frac{b}{b}$ .

$\frac{5 b}{a b} - \frac{3}{b} \cdot \frac{a}{a} + \frac{1}{a b}$

Step 3.2

Multiply $\frac{3}{b}$ by $\frac{a}{a}$ .

$\frac{5 b}{a b} - \frac{3 a}{b a} + \frac{1}{a b}$

Step 3.3

Reorder the factors of $b a$ .

$\frac{5 b}{a b} - \frac{3 a}{a b} + \frac{1}{a b}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 b - 3 a}{a b} + \frac{1}{a b}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 b - 3 a + 1}{a b}$

Step 6

Reorder $5 b$ and $- 3 a$ .

$\frac{- 3 a + 5 b + 1}{a b}$

Step 7

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

$\frac{- (3 a) + 5 b + 1}{a b}$

Step 8

Factor $- 1$ out of $5 b$ .

$\frac{- (3 a) - (- 5 b) + 1}{a b}$

Step 9

Factor $- 1$ out of $- (3 a) - (- 5 b)$ .

$\frac{- (3 a - 5 b) + 1}{a b}$

Step 10

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (3 a - 5 b) - 1 (- 1)}{a b}$

Step 11

Factor $- 1$ out of $- (3 a - 5 b) - 1 (- 1)$ .

$\frac{- (3 a - 5 b - 1)}{a b}$

Step 12

Simplify the expression.

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Step 12.1

Rewrite $- (3 a - 5 b - 1)$ as $- 1 (3 a - 5 b - 1)$ .

$\frac{- 1 (3 a - 5 b - 1)}{a b}$

Step 12.2

Move the negative in front of the fraction.

$- \frac{3 a - 5 b - 1}{a b}$