Precalculus Examples

$y - 73 = - 6 (x - (- 5))$

Step 1

Write $y = - 6 x + 43$ as a function.

$f (x) = - 6 x + 43$

Step 2

Solve for $y$ .

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Step 2.1

Simplify $- 6 (x - (- 5))$ .

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Step 2.1.1

Rewrite.

$y - 73 = 0 + 0 - 6 (x - (- 5))$

Step 2.1.2

Multiply $- 1$ by $- 5$ .

$y - 73 = - 6 (x + 5)$

Step 2.1.3

Apply the distributive property.

$y - 73 = - 6 x - 6 \cdot 5$

Step 2.1.4

Multiply $- 6$ by $5$ .

$y - 73 = - 6 x - 30$

Step 2.2

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

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Step 2.2.1

Add $73$ to both sides of the equation.

$y = - 6 x - 30 + 73$

Step 2.2.2

Add $- 30$ and $73$ .

$y = - 6 x + 43$

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.

$f (x + h) = - 6 (x + h) + 43$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Apply the distributive property.

$f (x + h) = - 6 x - 6 h + 43$

Step 4.1.2.2

The final answer is $- 6 x - 6 h + 43$ .

$- 6 x - 6 h + 43$

Step 4.2

Reorder $- 6 x$ and $- 6 h$ .

$- 6 h - 6 x + 43$

Step 4.3

Find the components of the definition.

$f (x + h) = - 6 h - 6 x + 43$

$f (x) = - 6 x + 43$

$f (x + h) = - 6 h - 6 x + 43$

$f (x) = - 6 x + 43$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 6 h - 6 x + 43 - (- 6 x + 43)}{h}$

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.

$\frac{- 6 h - 6 x + 43 - (- 6 x) - 1 \cdot 43}{h}$

Step 6.1.2

Multiply $- 6$ by $- 1$ .

$\frac{- 6 h - 6 x + 43 + 6 x - 1 \cdot 43}{h}$

Step 6.1.3

Multiply $- 1$ by $43$ .

$\frac{- 6 h - 6 x + 43 + 6 x - 43}{h}$

Step 6.1.4

Add $- 6 x$ and $6 x$ .

$\frac{- 6 h + 0 + 43 - 43}{h}$

Step 6.1.5

Add $- 6 h$ and $0$ .

$\frac{- 6 h + 43 - 43}{h}$

Step 6.1.6

Subtract $43$ from $43$ .

$\frac{- 6 h + 0}{h}$

Step 6.1.7

Add $- 6 h$ and $0$ .

$\frac{- 6 h}{h}$

Step 6.2

Cancel the common factor of $h$ .

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Step 6.2.1

Cancel the common factor.

$\frac{- 6 h}{h}$

Step 6.2.2

Divide $- 6$ by $1$ .

$- 6$

Step 7