Basic Math Examples

\frac{a + b}{6 a - b} - \frac{7 a}{b - 6 a}

$\frac{a + b}{6 a - b} - \frac{7 a}{b - 6 a}$

Step 1

Factor

- 1

$- 1$ out of

b

$b$ .

\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b) - 6 a}

$\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b) - 6 a}$

Step 2

Factor

- 1

$- 1$ out of

- 6 a

$- 6 a$ .

\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b) - (6 a)}

$\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b) - (6 a)}$

Step 3

Factor

- 1

$- 1$ out of

- 1 (- b) - (6 a)

$- 1 (- b) - (6 a)$ .

\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b + 6 a)}

$\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (- b + 6 a)}$

Step 4

Reorder terms.

\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (6 a - b)}

$\frac{a + b}{6 a - b} - \frac{7 a}{- 1 (6 a - b)}$

Step 5

To write

\frac{a + b}{6 a - b}

$\frac{a + b}{6 a - b}$ as a fraction with a common denominator, multiply by

\frac{- 1}{- 1}

$\frac{- 1}{- 1}$ .

\frac{a + b}{6 a - b} \cdot \frac{- 1}{- 1} - \frac{7 a}{- 1 (6 a - b)}

$\frac{a + b}{6 a - b} \cdot \frac{- 1}{- 1} - \frac{7 a}{- 1 (6 a - b)}$

Step 6

Write each expression with a common denominator of

(6 a - b) \cdot - 1

$(6 a - b) \cdot - 1$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 6.1

Multiply

\frac{a + b}{6 a - b}

$\frac{a + b}{6 a - b}$ by

\frac{- 1}{- 1}

$\frac{- 1}{- 1}$ .

\frac{(a + b) \cdot - 1}{(6 a - b) \cdot - 1} - \frac{7 a}{- 1 (6 a - b)}

$\frac{(a + b) \cdot - 1}{(6 a - b) \cdot - 1} - \frac{7 a}{- 1 (6 a - b)}$

Step 6.2

Reorder the factors of

(6 a - b) \cdot - 1

$(6 a - b) \cdot - 1$ .

\frac{(a + b) \cdot - 1}{- (6 a - b)} - \frac{7 a}{- 1 (6 a - b)}

$\frac{(a + b) \cdot - 1}{- (6 a - b)} - \frac{7 a}{- 1 (6 a - b)}$

\frac{(a + b) \cdot - 1}{- (6 a - b)} - \frac{7 a}{- 1 (6 a - b)}

$\frac{(a + b) \cdot - 1}{- (6 a - b)} - \frac{7 a}{- 1 (6 a - b)}$

Step 7

Combine the numerators over the common denominator.

\frac{(a + b) \cdot - 1 - 7 a}{- (6 a - b)}

$\frac{(a + b) \cdot - 1 - 7 a}{- (6 a - b)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

\frac{a \cdot - 1 + b \cdot - 1 - 7 a}{- (6 a - b)}

$\frac{a \cdot - 1 + b \cdot - 1 - 7 a}{- (6 a - b)}$

Step 8.2

Move

- 1

$- 1$ to the left of

a

$a$ .

\frac{- 1 \cdot a + b \cdot - 1 - 7 a}{- (6 a - b)}

$\frac{- 1 \cdot a + b \cdot - 1 - 7 a}{- (6 a - b)}$

Step 8.3

Move

- 1

$- 1$ to the left of

b

$b$ .

\frac{- 1 \cdot a - 1 \cdot b - 7 a}{- (6 a - b)}

$\frac{- 1 \cdot a - 1 \cdot b - 7 a}{- (6 a - b)}$

Step 8.4

Simplify each term.

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Step 8.4.1

Rewrite

$- 1 a$ as

$- a$ .

$\frac{- a - 1 \cdot b - 7 a}{- (6 a - b)}$

Step 8.4.2

Rewrite

$- 1 b$ as

$- b$ .

$\frac{- a - b - 7 a}{- (6 a - b)}$

Step 8.5

Subtract

$7 a$ from

$- a$ .

$\frac{- 8 a - b}{- (6 a - b)}$

Step 9

Simplify with factoring out.

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Step 9.1

Move the negative in front of the fraction.

$- \frac{- 8 a - b}{6 a - b}$

Step 9.2

Factor

$- 1$ out of

$- 8 a$ .

$- \frac{- (8 a) - b}{6 a - b}$

Step 9.3

Factor

$- 1$ out of

$- b$ .

$- \frac{- (8 a) - (b)}{6 a - b}$

Step 9.4

Factor

$- 1$ out of

$- (8 a) - (b)$ .

$- \frac{- (8 a + b)}{6 a - b}$

Step 9.5

Simplify the expression.

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Step 9.5.1

Rewrite

$- (8 a + b)$ as

$- 1 (8 a + b)$ .

$- \frac{- 1 (8 a + b)}{6 a - b}$

Step 9.5.2

Move the negative in front of the fraction.

$- - \frac{8 a + b}{6 a - b}$

Step 9.5.3

Multiply

$- 1$ by

$- 1$ .

$1 \frac{8 a + b}{6 a - b}$

Step 9.5.4

Multiply

$\frac{8 a + b}{6 a - b}$ by

$1$ .

$\frac{8 a + b}{6 a - b}$