Precalculus Examples

$- 5 - \frac{7}{8} x = y$

Step 1

Write $y = - 5 - \frac{7 x}{8}$ as a function.

$f (x) = - 5 - \frac{7 x}{8}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y = - 5 - \frac{7}{8} x$ .

$y = - 5 - \frac{7}{8} x$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $x$ and $\frac{7}{8}$ .

$y = - 5 - \frac{x \cdot 7}{8}$

Step 2.2.2

Move $7$ to the left of $x$ .

$y = - 5 - \frac{7 x}{8}$

Step 3

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 4

Find the components of the definition.

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Step 4.1

Evaluate the function at $x = x + h$ .

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Step 4.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - 5 - \frac{7 (x + h)}{8}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (x + h) = - 5 \cdot \frac{8}{8} - \frac{7 (x + h)}{8}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $- 5$ and $\frac{8}{8}$ .

$f (x + h) = \frac{- 5 \cdot 8}{8} - \frac{7 (x + h)}{8}$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{- 5 \cdot 8 - 7 (x + h)}{8}$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Factor $- 1$ out of $- 5 \cdot 8 - 7 (x + h)$ .

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Step 4.1.2.3.1.1

Reorder the expression.

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Step 4.1.2.3.1.1.1

Reorder $x$ and $h$ .

$f (x + h) = \frac{- 5 \cdot 8 - 7 (h + x)}{8}$

Step 4.1.2.3.1.1.2

Reorder $- 5 \cdot 8$ and $- 7 (h + x)$ .

$f (x + h) = \frac{- 7 (h + x) - 5 \cdot 8}{8}$

Step 4.1.2.3.1.2

Factor $- 1$ out of $- 7 (h + x)$ .

$f (x + h) = \frac{- (7 (h + x)) - 5 \cdot 8}{8}$

Step 4.1.2.3.1.3

Factor $- 1$ out of $- 5 \cdot 8$ .

$f (x + h) = \frac{- (7 (h + x)) - (5 \cdot 8)}{8}$

Step 4.1.2.3.1.4

Factor $- 1$ out of $- (7 (h + x)) - (5 \cdot 8)$ .

$f (x + h) = \frac{- (7 (h + x) + 5 \cdot 8)}{8}$

Step 4.1.2.3.2

Apply the distributive property.

$f (x + h) = \frac{- (7 h + 7 x + 5 \cdot 8)}{8}$

Step 4.1.2.3.3

Multiply $5$ by $8$ .

$f (x + h) = \frac{- (7 h + 7 x + 40)}{8}$

Step 4.1.2.4

Move the negative in front of the fraction.

$f (x + h) = - \frac{7 h + 7 x + 40}{8}$

Step 4.1.2.5

The final answer is $- \frac{7 h + 7 x + 40}{8}$ .

$- \frac{7 h + 7 x + 40}{8}$

Step 4.2

Find the components of the definition.

$f (x + h) = - \frac{7 h + 7 x + 40}{8}$

$f (x) = - 5 - \frac{7 x}{8}$

$f (x + h) = - \frac{7 h + 7 x + 40}{8}$

$f (x) = - 5 - \frac{7 x}{8}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{7 h + 7 x + 40}{8} - (- 5 - \frac{7 x}{8})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$\frac{- \frac{7 h + 7 x + 40}{8} - - 5 - - \frac{7 x}{8}}{h}$

Step 6.1.2

Multiply $- 1$ by $- 5$ .

$\frac{- \frac{7 h + 7 x + 40}{8} + 5 - - \frac{7 x}{8}}{h}$

Step 6.1.3

Multiply $- - \frac{7 x}{8}$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{7 h + 7 x + 40}{8} + 5 + 1 \frac{7 x}{8}}{h}$

Step 6.1.3.2

Multiply $\frac{7 x}{8}$ by $1$ .

$\frac{- \frac{7 h + 7 x + 40}{8} + 5 + \frac{7 x}{8}}{h}$

Step 6.1.4

To write $5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{- \frac{7 h + 7 x + 40}{8} + 5 \cdot \frac{8}{8} + \frac{7 x}{8}}{h}$

Step 6.1.5

Combine $5$ and $\frac{8}{8}$ .

$\frac{- \frac{7 h + 7 x + 40}{8} + \frac{5 \cdot 8}{8} + \frac{7 x}{8}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{- (7 h + 7 x + 40) + 5 \cdot 8}{8} + \frac{7 x}{8}}{h}$

Step 6.1.7

Combine the numerators over the common denominator.

$\frac{\frac{- (7 h + 7 x + 40) + 5 \cdot 8 + 7 x}{8}}{h}$

Step 6.1.8

Simplify each term.

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Step 6.1.8.1

Apply the distributive property.

$\frac{\frac{- (7 h) - (7 x) - 1 \cdot 40 + 5 \cdot 8 + 7 x}{8}}{h}$

Step 6.1.8.2

Simplify.

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Step 6.1.8.2.1

Multiply $7$ by $- 1$ .

$\frac{\frac{- 7 h - (7 x) - 1 \cdot 40 + 5 \cdot 8 + 7 x}{8}}{h}$

Step 6.1.8.2.2

Multiply $7$ by $- 1$ .

$\frac{\frac{- 7 h - 7 x - 1 \cdot 40 + 5 \cdot 8 + 7 x}{8}}{h}$

Step 6.1.8.2.3

Multiply $- 1$ by $40$ .

$\frac{\frac{- 7 h - 7 x - 40 + 5 \cdot 8 + 7 x}{8}}{h}$

Step 6.1.8.3

Multiply $5$ by $8$ .

$\frac{\frac{- 7 h - 7 x - 40 + 40 + 7 x}{8}}{h}$

Step 6.1.9

Combine the opposite terms in $- 7 h - 7 x - 40 + 40 + 7 x$ .

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Step 6.1.9.1

Add $- 40$ and $40$ .

$\frac{\frac{- 7 h - 7 x + 0 + 7 x}{8}}{h}$

Step 6.1.9.2

Add $- 7 h - 7 x$ and $0$ .

$\frac{\frac{- 7 h - 7 x + 7 x}{8}}{h}$

Step 6.1.9.3

Add $- 7 x$ and $7 x$ .

$\frac{\frac{- 7 h + 0}{8}}{h}$

Step 6.1.9.4

Add $- 7 h$ and $0$ .

$\frac{\frac{- 7 h}{8}}{h}$

Step 6.1.10

Move the negative in front of the fraction.

$\frac{- \frac{7 h}{8}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{7 h}{8} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{7 h}{8}$ into the numerator.

$\frac{- 7 h}{8} \cdot \frac{1}{h}$

Step 6.3.2

Factor $h$ out of $- 7 h$ .

$\frac{h \cdot - 7}{8} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$\frac{h \cdot - 7}{8} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$\frac{- 7}{8}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{7}{8}$

Step 7