Precalculus Examples

$y - \frac{2}{7} x = 5$

Step 1

Write $y = 5 + \frac{2 x}{7}$ as a function.

$f (x) = 5 + \frac{2 x}{7}$

Step 2

Solve for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{2}{7}$ .

$y - \frac{x \cdot 2}{7} = 5$

Step 2.1.2

Move $2$ to the left of $x$ .

$y - \frac{2 x}{7} = 5$

Step 2.2

Add $\frac{2 x}{7}$ to both sides of the equation.

$y = 5 + \frac{2 x}{7}$

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.

$f (x + h) = 5 + \frac{2 (x + h)}{7}$

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 $\frac{7}{7}$ .

$f (x + h) = 5 \cdot \frac{7}{7} + \frac{2 (x + h)}{7}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $5$ and $\frac{7}{7}$ .

$f (x + h) = \frac{5 \cdot 7}{7} + \frac{2 (x + h)}{7}$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{5 \cdot 7 + 2 (x + h)}{7}$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Multiply $5$ by $7$ .

$f (x + h) = \frac{35 + 2 (x + h)}{7}$

Step 4.1.2.3.2

Apply the distributive property.

$f (x + h) = \frac{35 + 2 x + 2 h}{7}$

Step 4.1.2.4

The final answer is $\frac{35 + 2 x + 2 h}{7}$ .

$\frac{35 + 2 x + 2 h}{7}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{35 + 2 x + 2 h}{7}$

$f (x) = 5 + \frac{2 x}{7}$

$f (x + h) = \frac{35 + 2 x + 2 h}{7}$

$f (x) = 5 + \frac{2 x}{7}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{35 + 2 x + 2 h}{7} - (5 + \frac{2 x}{7})}{h}$

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.

$\frac{\frac{35 + 2 x + 2 h}{7} - 1 \cdot 5 - \frac{2 x}{7}}{h}$

Step 6.1.2

Multiply $- 1$ by $5$ .

$\frac{\frac{35 + 2 x + 2 h}{7} - 5 - \frac{2 x}{7}}{h}$

Step 6.1.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{\frac{35 + 2 x + 2 h}{7} - 5 \cdot \frac{7}{7} - \frac{2 x}{7}}{h}$

Step 6.1.4

Combine $- 5$ and $\frac{7}{7}$ .

$\frac{\frac{35 + 2 x + 2 h}{7} + \frac{- 5 \cdot 7}{7} - \frac{2 x}{7}}{h}$

Step 6.1.5

Combine the numerators over the common denominator.

$\frac{\frac{35 + 2 x + 2 h - 5 \cdot 7}{7} - \frac{2 x}{7}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{35 + 2 x + 2 h - 5 \cdot 7 - 2 x}{7}}{h}$

Step 6.1.7

Rewrite $\frac{35 + 2 x + 2 h - 5 \cdot 7 - 2 x}{7}$ in a factored form.

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Step 6.1.7.1

Multiply $- 5$ by $7$ .

$\frac{\frac{35 + 2 x + 2 h - 35 - 2 x}{7}}{h}$

Step 6.1.7.2

Subtract $35$ from $35$ .

$\frac{\frac{2 x + 2 h + 0 - 2 x}{7}}{h}$

Step 6.1.7.3

Add $2 x$ and $0$ .

$\frac{\frac{2 x + 2 h - 2 x}{7}}{h}$

Step 6.1.7.4

Subtract $2 x$ from $2 x$ .

$\frac{\frac{2 h + 0}{7}}{h}$

Step 6.1.7.5

Add $2 h$ and $0$ .

$\frac{\frac{2 h}{7}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 h}{7} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $2 h$ .

$\frac{h \cdot 2}{7} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 2}{7} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{2}{7}$

Step 7