Basic Math Examples

$\frac{6 (z - 4)}{6 z - 7} - \frac{7 (z + 6)}{6 z - 7} - \frac{z - 4}{7 - 6 z}$

Step 1

Combine the numerators over the common denominator.

$\frac{6 (z - 4) - 7 (z + 6)}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{6 z + 6 \cdot - 4 - 7 (z + 6)}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 2.2

Multiply $6$ by $- 4$ .

$\frac{6 z - 24 - 7 (z + 6)}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 2.3

Apply the distributive property.

$\frac{6 z - 24 - 7 z - 7 \cdot 6}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 2.4

Multiply $- 7$ by $6$ .

$\frac{6 z - 24 - 7 z - 42}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 3

Simplify terms.

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Step 3.1

Subtract $7 z$ from $6 z$ .

$\frac{- z - 24 - 42}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $42$ from $- 24$ .

$\frac{- z - 66}{6 z - 7} + \frac{- (z - 4)}{- 6 z + 7}$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{- z - 66}{6 z - 7} - \frac{z - 4}{- 6 z + 7}$

Step 3.3

Factor $- 1$ out of $- 6 z$ .

$\frac{- z - 66}{6 z - 7} - \frac{z - 4}{- (6 z) + 7}$

Step 3.4

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- z - 66}{6 z - 7} - \frac{z - 4}{- (6 z) - 1 (- 7)}$

Step 3.5

Factor $- 1$ out of $- (6 z) - 1 (- 7)$ .

$\frac{- z - 66}{6 z - 7} - \frac{z - 4}{- (6 z - 7)}$

Step 4

To write $\frac{- z - 66}{6 z - 7}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{- z - 66}{6 z - 7} \cdot \frac{- 1}{- 1} - \frac{z - 4}{- (6 z - 7)}$

Step 5

Write each expression with a common denominator of $(6 z - 7) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{- z - 66}{6 z - 7}$ by $\frac{- 1}{- 1}$ .

$\frac{(- z - 66) \cdot - 1}{(6 z - 7) \cdot - 1} - \frac{z - 4}{- (6 z - 7)}$

Step 5.2

Reorder the factors of $(6 z - 7) \cdot - 1$ .

$\frac{(- z - 66) \cdot - 1}{- (6 z - 7)} - \frac{z - 4}{- (6 z - 7)}$

Step 6

Combine the numerators over the common denominator.

$\frac{(- z - 66) \cdot - 1 - (z - 4)}{- (6 z - 7)}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{- z \cdot - 1 - 66 \cdot - 1 - (z - 4)}{- (6 z - 7)}$

Step 7.2

Multiply $- z \cdot - 1$ .

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Step 7.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1 z - 66 \cdot - 1 - (z - 4)}{- (6 z - 7)}$

Step 7.2.2

Multiply $z$ by $1$ .

$\frac{z - 66 \cdot - 1 - (z - 4)}{- (6 z - 7)}$

Step 7.3

Multiply $- 66$ by $- 1$ .

$\frac{z + 66 - (z - 4)}{- (6 z - 7)}$

Step 7.4

Apply the distributive property.

$\frac{z + 66 - z - - 4}{- (6 z - 7)}$

Step 7.5

Multiply $- 1$ by $- 4$ .

$\frac{z + 66 - z + 4}{- (6 z - 7)}$

Step 7.6

Subtract $z$ from $z$ .

$\frac{0 + 66 + 4}{- (6 z - 7)}$

Step 7.7

Add $0$ and $66$ .

$\frac{66 + 4}{- (6 z - 7)}$

Step 7.8

Add $66$ and $4$ .

$\frac{70}{- (6 z - 7)}$

Step 8

Move the negative in front of the fraction.

$- \frac{70}{6 z - 7}$