Precalculus Examples

$y + \frac{2}{3} x = 1$

Step 1

Write $y = 1 - \frac{2 x}{3}$ as a function.

$f (x) = 1 - \frac{2 x}{3}$

Step 2

Solve for $y$ .

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Step 2.1

Combine $\frac{2}{3}$ and $x$ .

$y + \frac{2 x}{3} = 1$

Step 2.2

Subtract $\frac{2 x}{3}$ from both sides of the equation.

$y = 1 - \frac{2 x}{3}$

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.

$f (x + h) = 1 - \frac{2 (x + h)}{3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Combine into one fraction.

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Step 4.1.2.1.1

Write $1$ as a fraction with a common denominator.

$f (x + h) = \frac{3}{3} - \frac{2 (x + h)}{3}$

Step 4.1.2.1.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{3 - 2 (x + h)}{3}$

Step 4.1.2.2

Apply the distributive property.

$f (x + h) = \frac{3 - 2 x - 2 h}{3}$

Step 4.1.2.3

The final answer is $\frac{3 - 2 x - 2 h}{3}$ .

$\frac{3 - 2 x - 2 h}{3}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{3 - 2 x - 2 h}{3}$

$f (x) = 1 - \frac{2 x}{3}$

$f (x + h) = \frac{3 - 2 x - 2 h}{3}$

$f (x) = 1 - \frac{2 x}{3}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{3 - 2 x - 2 h}{3} - (1 - \frac{2 x}{3})}{h}$

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.

$\frac{\frac{3 - 2 x - 2 h}{3} - 1 \cdot 1 - - \frac{2 x}{3}}{h}$

Step 6.1.2

Multiply $- 1$ by $1$ .

$\frac{\frac{3 - 2 x - 2 h}{3} - 1 - - \frac{2 x}{3}}{h}$

Step 6.1.3

Multiply $- - \frac{2 x}{3}$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{3 - 2 x - 2 h}{3} - 1 + 1 \frac{2 x}{3}}{h}$

Step 6.1.3.2

Multiply $\frac{2 x}{3}$ by $1$ .

$\frac{\frac{3 - 2 x - 2 h}{3} - 1 + \frac{2 x}{3}}{h}$

Step 6.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{3 - 2 x - 2 h}{3} - 1 \cdot \frac{3}{3} + \frac{2 x}{3}}{h}$

Step 6.1.5

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{\frac{3 - 2 x - 2 h}{3} + \frac{- 1 \cdot 3}{3} + \frac{2 x}{3}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{3 - 2 x - 2 h - 1 \cdot 3}{3} + \frac{2 x}{3}}{h}$

Step 6.1.7

Combine the numerators over the common denominator.

$\frac{\frac{3 - 2 x - 2 h - 1 \cdot 3 + 2 x}{3}}{h}$

Step 6.1.8

Rewrite $\frac{3 - 2 x - 2 h - 1 \cdot 3 + 2 x}{3}$ in a factored form.

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Step 6.1.8.1

Multiply $- 1$ by $3$ .

$\frac{\frac{3 - 2 x - 2 h - 3 + 2 x}{3}}{h}$

Step 6.1.8.2

Subtract $3$ from $3$ .

$\frac{\frac{- 2 x - 2 h + 0 + 2 x}{3}}{h}$

Step 6.1.8.3

Add $- 2 x$ and $0$ .

$\frac{\frac{- 2 x - 2 h + 2 x}{3}}{h}$

Step 6.1.8.4

Add $- 2 x$ and $2 x$ .

$\frac{\frac{- 2 h + 0}{3}}{h}$

Step 6.1.8.5

Add $- 2 h$ and $0$ .

$\frac{\frac{- 2 h}{3}}{h}$

Step 6.1.9

Move the negative in front of the fraction.

$\frac{- \frac{2 h}{3}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2 h}{3} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{2 h}{3}$ into the numerator.

$\frac{- 2 h}{3} \cdot \frac{1}{h}$

Step 6.3.2

Factor $h$ out of $- 2 h$ .

$\frac{h \cdot - 2}{3} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$\frac{h \cdot - 2}{3} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$\frac{- 2}{3}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 7