Precalculus Examples

$25 x - 4 y = 50$

Step 1

Write $y = - \frac{25}{2} + \frac{25 x}{4}$ as a function.

$f (x) = - \frac{25}{2} + \frac{25 x}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Subtract $25 x$ from both sides of the equation.

$- 4 y = 50 - 25 x$

Step 2.2

Divide each term in $- 4 y = 50 - 25 x$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 y = 50 - 25 x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{50}{- 4} + \frac{- 25 x}{- 4}$

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.

$\frac{- 4 y}{- 4} = \frac{50}{- 4} + \frac{- 25 x}{- 4}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{50}{- 4} + \frac{- 25 x}{- 4}$

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

Cancel the common factor of $50$ and $- 4$ .

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Step 2.2.3.1.1.1

Factor $2$ out of $50$ .

$y = \frac{2 (25)}{- 4} + \frac{- 25 x}{- 4}$

Step 2.2.3.1.1.2

Cancel the common factors.

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Step 2.2.3.1.1.2.1

Factor $2$ out of $- 4$ .

$y = \frac{2 \cdot 25}{2 \cdot - 2} + \frac{- 25 x}{- 4}$

Step 2.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 25}{2 \cdot - 2} + \frac{- 25 x}{- 4}$

Step 2.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{25}{- 2} + \frac{- 25 x}{- 4}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{25}{2} + \frac{- 25 x}{- 4}$

Step 2.2.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{25}{2} + \frac{25 x}{4}$

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.

$f (x + h) = - \frac{25}{2} + \frac{25 (x + h)}{4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

To write $- \frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (x + h) = - \frac{25}{2} \cdot \frac{2}{2} + \frac{25 (x + h)}{4}$

Step 4.1.2.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.2.1

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (x + h) = - \frac{25 \cdot 2}{2 \cdot 2} + \frac{25 (x + h)}{4}$

Step 4.1.2.2.2

Multiply $2$ by $2$ .

$f (x + h) = - \frac{25 \cdot 2}{4} + \frac{25 (x + h)}{4}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$f (x + h) = \frac{- 25 \cdot 2 + 25 (x + h)}{4}$

Step 4.1.2.4

Simplify the numerator.

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Step 4.1.2.4.1

Factor $25$ out of $- 25 \cdot 2 + 25 (x + h)$ .

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Step 4.1.2.4.1.1

Factor $25$ out of $- 25 \cdot 2$ .

$f (x + h) = \frac{25 (- 1 \cdot 2) + 25 (x + h)}{4}$

Step 4.1.2.4.1.2

Factor $25$ out of $25 (- 1 \cdot 2) + 25 (x + h)$ .

$f (x + h) = \frac{25 (- 1 \cdot 2 + x + h)}{4}$

Step 4.1.2.4.2

Multiply $- 1$ by $2$ .

$f (x + h) = \frac{25 (- 2 + x + h)}{4}$

Step 4.1.2.5

Simplify with factoring out.

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Step 4.1.2.5.1

Rewrite $- 2$ as $- 1 (2)$ .

$f (x + h) = \frac{25 (- 1 \cdot 2 + x + h)}{4}$

Step 4.1.2.5.2

Factor $- 1$ out of $x$ .

$f (x + h) = \frac{25 (- 1 \cdot 2 - 1 (- x) + h)}{4}$

Step 4.1.2.5.3

Factor $- 1$ out of $- 1 (2) - 1 (- x)$ .

$f (x + h) = \frac{25 (- 1 (2 - x) + h)}{4}$

Step 4.1.2.5.4

Factor $- 1$ out of $h$ .

$f (x + h) = \frac{25 (- 1 (2 - x) - 1 (- h))}{4}$

Step 4.1.2.5.5

Factor $- 1$ out of $- 1 (2 - x) - 1 (- h)$ .

$f (x + h) = \frac{25 (- 1 (2 - x - h))}{4}$

Step 4.1.2.5.6

Move the negative in front of the fraction.

$f (x + h) = - \frac{25 (2 - x - h)}{4}$

Step 4.1.2.6

The final answer is $- \frac{25 (2 - x - h)}{4}$ .

$- \frac{25 (2 - x - h)}{4}$

Step 4.2

Find the components of the definition.

$f (x + h) = - \frac{25 (2 - x - h)}{4}$

$f (x) = - \frac{25}{2} + \frac{25 x}{4}$

$f (x + h) = - \frac{25 (2 - x - h)}{4}$

$f (x) = - \frac{25}{2} + \frac{25 x}{4}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{25 (2 - x - h)}{4} - (- \frac{25}{2} + \frac{25 x}{4})}{h}$

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.

$\frac{- \frac{25 (2 - x - h)}{4} - - \frac{25}{2} - \frac{25 x}{4}}{h}$

Step 6.1.2

Multiply $- - \frac{25}{2}$ .

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{25 (2 - x - h)}{4} + 1 (\frac{25}{2}) - \frac{25 x}{4}}{h}$

Step 6.1.2.2

Multiply $\frac{25}{2}$ by $1$ .

$\frac{- \frac{25 (2 - x - h)}{4} + \frac{25}{2} - \frac{25 x}{4}}{h}$

Step 6.1.3

To write $\frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{25 (2 - x - h)}{4} + \frac{25}{2} \cdot \frac{2}{2} - \frac{25 x}{4}}{h}$

Step 6.1.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.4.1

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$\frac{- \frac{25 (2 - x - h)}{4} + \frac{25 \cdot 2}{2 \cdot 2} - \frac{25 x}{4}}{h}$

Step 6.1.4.2

Multiply $2$ by $2$ .

$\frac{- \frac{25 (2 - x - h)}{4} + \frac{25 \cdot 2}{4} - \frac{25 x}{4}}{h}$

Step 6.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- 25 (2 - x - h) + 25 \cdot 2}{4} - \frac{25 x}{4}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{- 25 (2 - x - h) + 25 \cdot 2 - 25 x}{4}}{h}$

Step 6.1.7

Simplify each term.

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Step 6.1.7.1

Apply the distributive property.

$\frac{\frac{- 25 \cdot 2 - 25 (- x) - 25 (- h) + 25 \cdot 2 - 25 x}{4}}{h}$

Step 6.1.7.2

Simplify.

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Step 6.1.7.2.1

Multiply $- 25$ by $2$ .

$\frac{\frac{- 50 - 25 (- x) - 25 (- h) + 25 \cdot 2 - 25 x}{4}}{h}$

Step 6.1.7.2.2

Multiply $- 1$ by $- 25$ .

$\frac{\frac{- 50 + 25 x - 25 (- h) + 25 \cdot 2 - 25 x}{4}}{h}$

Step 6.1.7.2.3

Multiply $- 1$ by $- 25$ .

$\frac{\frac{- 50 + 25 x + 25 h + 25 \cdot 2 - 25 x}{4}}{h}$

Step 6.1.7.3

Multiply $25$ by $2$ .

$\frac{\frac{- 50 + 25 x + 25 h + 50 - 25 x}{4}}{h}$

Step 6.1.8

Combine the opposite terms in $- 50 + 25 x + 25 h + 50 - 25 x$ .

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Step 6.1.8.1

Add $- 50$ and $50$ .

$\frac{\frac{25 x + 25 h + 0 - 25 x}{4}}{h}$

Step 6.1.8.2

Add $25 x + 25 h$ and $0$ .

$\frac{\frac{25 x + 25 h - 25 x}{4}}{h}$

Step 6.1.8.3

Subtract $25 x$ from $25 x$ .

$\frac{\frac{25 h + 0}{4}}{h}$

Step 6.1.8.4

Add $25 h$ and $0$ .

$\frac{\frac{25 h}{4}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{25 h}{4} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $25 h$ .

$\frac{h \cdot 25}{4} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 25}{4} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{25}{4}$

Step 7