Algebra Examples

$\frac{1}{a + b} + \frac{2}{a} - \frac{3}{b}$

Step 1

To write $\frac{1}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{1}{a + b} \cdot \frac{a}{a} + \frac{2}{a} - \frac{3}{b}$

Step 2

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

$\frac{1}{a + b} \cdot \frac{a}{a} + \frac{2}{a} \cdot \frac{a + b}{a + b} - \frac{3}{b}$

Step 3

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

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Step 3.1

Multiply $\frac{1}{a + b}$ by $\frac{a}{a}$ .

$\frac{a}{(a + b) a} + \frac{2}{a} \cdot \frac{a + b}{a + b} - \frac{3}{b}$

Step 3.2

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

$\frac{a}{(a + b) a} + \frac{2 (a + b)}{a (a + b)} - \frac{3}{b}$

Step 3.3

Reorder the factors of $(a + b) a$ .

$\frac{a}{a (a + b)} + \frac{2 (a + b)}{a (a + b)} - \frac{3}{b}$

Step 4

Combine the numerators over the common denominator.

$\frac{a + 2 (a + b)}{a (a + b)} - \frac{3}{b}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{a + 2 a + 2 b}{a (a + b)} - \frac{3}{b}$

Step 5.2

Add $a$ and $2 a$ .

$\frac{3 a + 2 b}{a (a + b)} - \frac{3}{b}$

Step 6

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

$\frac{3 a + 2 b}{a (a + b)} \cdot \frac{b}{b} - \frac{3}{b}$

Step 7

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

$\frac{3 a + 2 b}{a (a + b)} \cdot \frac{b}{b} - \frac{3}{b} \cdot \frac{a (a + b)}{a (a + b)}$

Step 8

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

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Step 8.1

Multiply $\frac{3 a + 2 b}{a (a + b)}$ by $\frac{b}{b}$ .

$\frac{(3 a + 2 b) b}{a (a + b) b} - \frac{3}{b} \cdot \frac{a (a + b)}{a (a + b)}$

Step 8.2

Multiply $\frac{3}{b}$ by $\frac{a (a + b)}{a (a + b)}$ .

$\frac{(3 a + 2 b) b}{a (a + b) b} - \frac{3 (a (a + b))}{b (a (a + b))}$

Step 8.3

Reorder the factors of $a (a + b) b$ .

$\frac{(3 a + 2 b) b}{a b (a + b)} - \frac{3 (a (a + b))}{b (a (a + b))}$

Step 8.4

Reorder the factors of $b (a (a + b))$ .

$\frac{(3 a + 2 b) b}{a b (a + b)} - \frac{3 (a (a + b))}{a b (a + b)}$

Step 9

Combine the numerators over the common denominator.

$\frac{(3 a + 2 b) b - 3 (a (a + b))}{a b (a + b)}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{3 a b + 2 b \cdot b - 3 a (a + b)}{a b (a + b)}$

Step 10.2

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

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Step 10.2.1

Move $b$ .

$\frac{3 a b + 2 (b \cdot b) - 3 a (a + b)}{a b (a + b)}$

Step 10.2.2

Multiply $b$ by $b$ .

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

Step 10.3

Apply the distributive property.

$\frac{3 a b + 2 b^{2} - 3 a \cdot a - 3 a b}{a b (a + b)}$

Step 10.4

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

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Step 10.4.1

Move $a$ .

$\frac{3 a b + 2 b^{2} - 3 (a \cdot a) - 3 a b}{a b (a + b)}$

Step 10.4.2

Multiply $a$ by $a$ .

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

Step 10.5

Subtract $3 a b$ from $3 a b$ .

$\frac{2 b^{2} - 3 a^{2} + 0}{a b (a + b)}$

Step 10.6

Add $2 b^{2} - 3 a^{2}$ and $0$ .

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