Precalculus Examples

$2 y = 5 x + 10$

Step 1

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

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

Step 2

Divide each term in $2 y = 5 x + 10$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 y = 5 x + 10$ by $2$ .

$\frac{2 y}{2} = \frac{5 x}{2} + \frac{10}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{5 x}{2} + \frac{10}{2}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{2} + \frac{10}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $10$ by $2$ .

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

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.

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

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 $\frac{2}{2}$ .

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

Step 4.1.2.2

Combine $5$ and $\frac{2}{2}$ .

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

Step 4.1.2.3

Combine the numerators over the common denominator.

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

Step 4.1.2.4

Factor $5$ out of $5 (x + h) + 5 \cdot 2$ .

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Step 4.1.2.4.1

Factor $5$ out of $5 \cdot 2$ .

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

Step 4.1.2.4.2

Factor $5$ out of $5 (x + h) + 5 (2)$ .

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

Step 4.1.2.5

The final answer is $\frac{5 (x + h + 2)}{2}$ .

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

Step 4.2

Find the components of the definition.

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

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

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

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

Step 5

Plug in the components.

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

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.

$\frac{\frac{5 (x + h + 2)}{2} - \frac{5 x}{2} - 1 \cdot 5}{h}$

Step 6.1.2

Multiply $- 1$ by $5$ .

$\frac{\frac{5 (x + h + 2)}{2} - \frac{5 x}{2} - 5}{h}$

Step 6.1.3

Combine the numerators over the common denominator.

$\frac{\frac{5 (x + h + 2) - 5 x}{2} - 5}{h}$

Step 6.1.4

Simplify the numerator.

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Step 6.1.4.1

Factor $5$ out of $5 (x + h + 2) - 5 x$ .

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Step 6.1.4.1.1

Factor $5$ out of $- 5 x$ .

$\frac{\frac{5 (x + h + 2) + 5 (- x)}{2} - 5}{h}$

Step 6.1.4.1.2

Factor $5$ out of $5 (x + h + 2) + 5 (- x)$ .

$\frac{\frac{5 (x + h + 2 - x)}{2} - 5}{h}$

Step 6.1.4.2

Subtract $x$ from $x$ .

$\frac{\frac{5 (h + 2 + 0)}{2} - 5}{h}$

Step 6.1.4.3

Add $h + 2$ and $0$ .

$\frac{\frac{5 (h + 2)}{2} - 5}{h}$

Step 6.1.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{5 (h + 2)}{2} - 5 \cdot \frac{2}{2}}{h}$

Step 6.1.6

Combine $- 5$ and $\frac{2}{2}$ .

$\frac{\frac{5 (h + 2)}{2} + \frac{- 5 \cdot 2}{2}}{h}$

Step 6.1.7

Combine the numerators over the common denominator.

$\frac{\frac{5 (h + 2) - 5 \cdot 2}{2}}{h}$

Step 6.1.8

Simplify the numerator.

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Step 6.1.8.1

Factor $5$ out of $5 (h + 2) - 5 \cdot 2$ .

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Step 6.1.8.1.1

Factor $5$ out of $- 5 \cdot 2$ .

$\frac{\frac{5 (h + 2) + 5 (- 1 \cdot 2)}{2}}{h}$

Step 6.1.8.1.2

Factor $5$ out of $5 (h + 2) + 5 (- 1 \cdot 2)$ .

$\frac{\frac{5 (h + 2 - 1 \cdot 2)}{2}}{h}$

Step 6.1.8.2

Multiply $- 1$ by $2$ .

$\frac{\frac{5 (h + 2 - 2)}{2}}{h}$

Step 6.1.8.3

Subtract $2$ from $2$ .

$\frac{\frac{5 (h + 0)}{2}}{h}$

Step 6.1.8.4

Add $h$ and $0$ .

$\frac{\frac{5 h}{2}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 h}{2} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $5 h$ .

$\frac{h \cdot 5}{2} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 5}{2} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{5}{2}$

Step 7