Precalculus Examples

$x + 4 y = 20$

Step 1

Write $y = 5 - \frac{x}{4}$ as a function.

$f (x) = 5 - \frac{x}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Subtract $x$ from both sides of the equation.

$4 y = 20 - x$

Step 2.2

Divide each term in $4 y = 20 - x$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 y = 20 - x$ by $4$ .

$\frac{4 y}{4} = \frac{20}{4} + \frac{- x}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{20}{4} + \frac{- x}{4}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{4} + \frac{- x}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $20$ by $4$ .

$y = 5 + \frac{- x}{4}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = 5 - \frac{x}{4}$

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.

$f (x + h) = 5 - \frac{x + h}{4}$

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 $\frac{4}{4}$ .

$f (x + h) = 5 \cdot \frac{4}{4} - \frac{x + h}{4}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $5$ and $\frac{4}{4}$ .

$f (x + h) = \frac{5 \cdot 4}{4} - \frac{x + h}{4}$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{5 \cdot 4 - (x + h)}{4}$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Multiply $5$ by $4$ .

$f (x + h) = \frac{20 - (x + h)}{4}$

Step 4.1.2.3.2

Apply the distributive property.

$f (x + h) = \frac{20 - x - h}{4}$

Step 4.1.2.4

The final answer is $\frac{20 - x - h}{4}$ .

$\frac{20 - x - h}{4}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{20 - x - h}{4}$

$f (x) = 5 - \frac{x}{4}$

$f (x + h) = \frac{20 - x - h}{4}$

$f (x) = 5 - \frac{x}{4}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{20 - x - h}{4} - (5 - \frac{x}{4})}{h}$

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.

$\frac{\frac{20 - x - h}{4} - 1 \cdot 5 - - \frac{x}{4}}{h}$

Step 6.1.2

Multiply $- 1$ by $5$ .

$\frac{\frac{20 - x - h}{4} - 5 - - \frac{x}{4}}{h}$

Step 6.1.3

Multiply $- - \frac{x}{4}$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{20 - x - h}{4} - 5 + 1 \frac{x}{4}}{h}$

Step 6.1.3.2

Multiply $\frac{x}{4}$ by $1$ .

$\frac{\frac{20 - x - h}{4} - 5 + \frac{x}{4}}{h}$

Step 6.1.4

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{20 - x - h}{4} - 5 \cdot \frac{4}{4} + \frac{x}{4}}{h}$

Step 6.1.5

Combine $- 5$ and $\frac{4}{4}$ .

$\frac{\frac{20 - x - h}{4} + \frac{- 5 \cdot 4}{4} + \frac{x}{4}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{20 - x - h - 5 \cdot 4}{4} + \frac{x}{4}}{h}$

Step 6.1.7

Combine the numerators over the common denominator.

$\frac{\frac{20 - x - h - 5 \cdot 4 + x}{4}}{h}$

Step 6.1.8

Rewrite $\frac{20 - x - h - 5 \cdot 4 + x}{4}$ in a factored form.

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Step 6.1.8.1

Multiply $- 5$ by $4$ .

$\frac{\frac{20 - x - h - 20 + x}{4}}{h}$

Step 6.1.8.2

Subtract $20$ from $20$ .

$\frac{\frac{- x - h + 0 + x}{4}}{h}$

Step 6.1.8.3

Add $- x$ and $0$ .

$\frac{\frac{- x - h + x}{4}}{h}$

Step 6.1.8.4

Add $- x$ and $x$ .

$\frac{\frac{- h + 0}{4}}{h}$

Step 6.1.8.5

Add $- h$ and $0$ .

$\frac{\frac{- h}{4}}{h}$

Step 6.1.9

Move the negative in front of the fraction.

$\frac{- \frac{h}{4}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{4} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{h}{4}$ into the numerator.

$\frac{- h}{4} \cdot \frac{1}{h}$

Step 6.3.2

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{4} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$\frac{h \cdot - 1}{4} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$\frac{- 1}{4}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7