Precalculus Examples

$9 x - 11 y + 2 = 0$

Step 1

Write $y = \frac{9 x}{11} + \frac{2}{11}$ as a function.

$f (x) = \frac{9 x}{11} + \frac{2}{11}$

Step 2

Solve for $y$ .

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Step 2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.1.1

Subtract $9 x$ from both sides of the equation.

$- 11 y + 2 = - 9 x$

Step 2.1.2

Subtract $2$ from both sides of the equation.

$- 11 y = - 9 x - 2$

Step 2.2

Divide each term in $- 11 y = - 9 x - 2$ by $- 11$ and simplify.

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Step 2.2.1

Divide each term in $- 11 y = - 9 x - 2$ by $- 11$ .

$\frac{- 11 y}{- 11} = \frac{- 9 x}{- 11} + \frac{- 2}{- 11}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 11$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 11 y}{- 11} = \frac{- 9 x}{- 11} + \frac{- 2}{- 11}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9 x}{- 11} + \frac{- 2}{- 11}$

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

Dividing two negative values results in a positive value.

$y = \frac{9 x}{11} + \frac{- 2}{- 11}$

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{9 x}{11} + \frac{2}{11}$

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.

$f (x + h) = \frac{9 (x + h)}{11} + \frac{2}{11}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Combine the numerators over the common denominator.

$f (x + h) = \frac{9 (x + h) + 2}{11}$

Step 4.1.2.2

Apply the distributive property.

$f (x + h) = \frac{9 x + 9 h + 2}{11}$

Step 4.1.2.3

The final answer is $\frac{9 x + 9 h + 2}{11}$ .

$\frac{9 x + 9 h + 2}{11}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{9 x + 9 h + 2}{11}$

$f (x) = \frac{9 x}{11} + \frac{2}{11}$

$f (x + h) = \frac{9 x + 9 h + 2}{11}$

$f (x) = \frac{9 x}{11} + \frac{2}{11}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{9 x + 9 h + 2}{11} - (\frac{9 x}{11} + \frac{2}{11})}{h}$

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.

$\frac{\frac{9 x + 9 h + 2}{11} - \frac{9 x}{11} - \frac{2}{11}}{h}$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{\frac{9 x + 9 h + 2 - 9 x}{11} - \frac{2}{11}}{h}$

Step 6.1.3

Combine the numerators over the common denominator.

$\frac{\frac{9 x + 9 h + 2 - 9 x - 2}{11}}{h}$

Step 6.1.4

Rewrite $\frac{9 x + 9 h + 2 - 9 x - 2}{11}$ in a factored form.

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Step 6.1.4.1

Subtract $9 x$ from $9 x$ .

$\frac{\frac{9 h + 2 + 0 - 2}{11}}{h}$

Step 6.1.4.2

Add $9 h$ and $0$ .

$\frac{\frac{9 h + 2 - 2}{11}}{h}$

Step 6.1.4.3

Subtract $2$ from $2$ .

$\frac{\frac{9 h + 0}{11}}{h}$

Step 6.1.4.4

Add $9 h$ and $0$ .

$\frac{\frac{9 h}{11}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 h}{11} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $9 h$ .

$\frac{h \cdot 9}{11} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 9}{11} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{9}{11}$

Step 7