Algebra Examples

$\frac{b - a}{a b} + \frac{c - b}{b c} - \frac{c - a}{a c}$

Step 1

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

$\frac{b - a}{a b} \cdot \frac{c}{c} + \frac{c - b}{b c} - \frac{c - a}{a c}$

Step 2

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

$\frac{b - a}{a b} \cdot \frac{c}{c} + \frac{c - b}{b c} \cdot \frac{a}{a} - \frac{c - a}{a c}$

Step 3

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

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Step 3.1

Multiply $\frac{b - a}{a b}$ by $\frac{c}{c}$ .

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

Step 3.2

Multiply $\frac{c - b}{b c}$ by $\frac{a}{a}$ .

$\frac{(b - a) c}{a b c} + \frac{(c - b) a}{b c a} - \frac{c - a}{a c}$

Step 3.3

Reorder the factors of $b c a$ .

$\frac{(b - a) c}{a b c} + \frac{(c - b) a}{a b c} - \frac{c - a}{a c}$

Step 4

Combine the numerators over the common denominator.

$\frac{(b - a) c + (c - b) a}{a b c} - \frac{c - a}{a c}$

Step 5

Simplify each term.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{b c - a c + (c - b) a}{a b c} - \frac{c - a}{a c}$

Step 5.1.2

Apply the distributive property.

$\frac{b c - a c + c a - b a}{a b c} - \frac{c - a}{a c}$

Step 5.1.3

Add $- a c$ and $c a$ .

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Step 5.1.3.1

Reorder $c$ and $a$ .

$\frac{b c - a c + a c - b a}{a b c} - \frac{c - a}{a c}$

Step 5.1.3.2

Add $- a c$ and $a c$ .

$\frac{b c + 0 - b a}{a b c} - \frac{c - a}{a c}$

Step 5.1.4

Add $b c$ and $0$ .

$\frac{b c - b a}{a b c} - \frac{c - a}{a c}$

Step 5.1.5

Rewrite $b c - b a$ in a factored form.

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Step 5.1.5.1

Factor $b$ out of $b c - b a$ .

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Step 5.1.5.1.1

Factor $b$ out of $b c$ .

$\frac{b (c) - b a}{a b c} - \frac{c - a}{a c}$

Step 5.1.5.1.2

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

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

Step 5.1.5.1.3

Factor $b$ out of $b (c) + b (- 1 a)$ .

$\frac{b (c - 1 a)}{a b c} - \frac{c - a}{a c}$

Step 5.1.5.2

Rewrite $- 1 a$ as $- a$ .

$\frac{b (c - a)}{a b c} - \frac{c - a}{a c}$

Step 5.2

Cancel the common factor of $b$ .

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Step 5.2.1

Cancel the common factor.

$\frac{b (c - a)}{a b c} - \frac{c - a}{a c}$

Step 5.2.2

Rewrite the expression.

$\frac{c - a}{a c} - \frac{c - a}{a c}$

Step 6

Subtract $\frac{c - a}{a c}$ from $\frac{c - a}{a c}$ .

$0$