Calculus Examples

$y = 6 - x$ , $x = \frac{{(y - 6)}^{2}}{10}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Replace all occurrences of $x$ with $\frac{{(y - 6)}^{2}}{10}$ in each equation.

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Step 1.1.1

Replace all occurrences of $x$ in $y = 6 - x$ with $\frac{{(y - 6)}^{2}}{10}$ .

$y = 6 - (\frac{{(y - 6)}^{2}}{10})$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Simplify $6 - (\frac{{(y - 6)}^{2}}{10})$ .

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Step 1.1.2.1.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y = 6 \cdot \frac{10}{10} - \frac{{(y - 6)}^{2}}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.2

Simplify terms.

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Step 1.1.2.1.2.1

Combine $6$ and $\frac{10}{10}$ .

$y = \frac{6 \cdot 10}{10} - \frac{{(y - 6)}^{2}}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{6 \cdot 10 - {(y - 6)}^{2}}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{6 \cdot 10 - {(y - 6)}^{2}}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3

Simplify the numerator.

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Step 1.1.2.1.3.1

Multiply $6$ by $10$ .

$y = \frac{60 - {(y - 6)}^{2}}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.2

Rewrite ${(y - 6)}^{2}$ as $(y - 6) (y - 6)$ .

$y = \frac{60 - ((y - 6) (y - 6))}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.3

Expand $(y - 6) (y - 6)$ using the FOIL Method.

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Step 1.1.2.1.3.3.1

Apply the distributive property.

$y = \frac{60 - (y (y - 6) - 6 (y - 6))}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.3.2

Apply the distributive property.

$y = \frac{60 - (y \cdot y + y \cdot - 6 - 6 (y - 6))}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.3.3

Apply the distributive property.

$y = \frac{60 - (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{60 - (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.4

Simplify and combine like terms.

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Step 1.1.2.1.3.4.1

Simplify each term.

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Step 1.1.2.1.3.4.1.1

Multiply $y$ by $y$ .

$y = \frac{60 - (y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.4.1.2

Move $- 6$ to the left of $y$ .

$y = \frac{60 - (y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.4.1.3

Multiply $- 6$ by $- 6$ .

$y = \frac{60 - (y^{2} - 6 y - 6 y + 36)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{60 - (y^{2} - 6 y - 6 y + 36)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.4.2

Subtract $6 y$ from $- 6 y$ .

$y = \frac{60 - (y^{2} - 12 y + 36)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{60 - (y^{2} - 12 y + 36)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.5

Apply the distributive property.

$y = \frac{60 - y^{2} - (- 12 y) - 1 \cdot 36}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.6

Simplify.

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Step 1.1.2.1.3.6.1

Multiply $- 12$ by $- 1$ .

$y = \frac{60 - y^{2} + 12 y - 1 \cdot 36}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.6.2

Multiply $- 1$ by $36$ .

$y = \frac{60 - y^{2} + 12 y - 36}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{60 - y^{2} + 12 y - 36}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.3.7

Subtract $36$ from $60$ .

$y = \frac{- y^{2} + 12 y + 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = \frac{- y^{2} + 12 y + 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4

Simplify with factoring out.

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Step 1.1.2.1.4.1

Factor $- 1$ out of $- y^{2}$ .

$y = \frac{- (y^{2}) + 12 y + 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.2

Factor $- 1$ out of $12 y$ .

$y = \frac{- (y^{2}) - (- 12 y) + 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.3

Factor $- 1$ out of $- (y^{2}) - (- 12 y)$ .

$y = \frac{- (y^{2} - 12 y) + 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.4

Rewrite $24$ as $- 1 (- 24)$ .

$y = \frac{- (y^{2} - 12 y) - 1 \cdot - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.5

Factor $- 1$ out of $- (y^{2} - 12 y) - 1 (- 24)$ .

$y = \frac{- (y^{2} - 12 y - 24)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.6

Simplify the expression.

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Step 1.1.2.1.4.6.1

Rewrite $- (y^{2} - 12 y - 24)$ as $- 1 (y^{2} - 12 y - 24)$ .

$y = \frac{- 1 (y^{2} - 12 y - 24)}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.1.2.1.4.6.2

Move the negative in front of the fraction.

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - \frac{y^{2} - 12 y - 24}{10}$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2

Solve for $y$ in $y = - \frac{y^{2} - 12 y - 24}{10}$ .

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Step 1.2.1

Multiply both sides by $10$ .

$y \cdot 10 = - \frac{y^{2} - 12 y - 24}{10} \cdot 10$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Move $10$ to the left of $y$ .

$10 y = - \frac{y^{2} - 12 y - 24}{10} \cdot 10$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - \frac{y^{2} - 12 y - 24}{10} \cdot 10$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Simplify $- \frac{y^{2} - 12 y - 24}{10} \cdot 10$ .

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Step 1.2.2.2.1.1

Cancel the common factor of $10$ .

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Step 1.2.2.2.1.1.1

Move the leading negative in $- \frac{y^{2} - 12 y - 24}{10}$ into the numerator.

$10 y = \frac{- (y^{2} - 12 y - 24)}{10} \cdot 10$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2.1.1.2

Cancel the common factor.

$10 y = \frac{- (y^{2} - 12 y - 24)}{10} \cdot 10$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2.1.1.3

Rewrite the expression.

$10 y = - (y^{2} - 12 y - 24)$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - (y^{2} - 12 y - 24)$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2.1.2

Apply the distributive property.

$10 y = - y^{2} - (- 12 y) + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2.1.3

Simplify.

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Step 1.2.2.2.1.3.1

Multiply $- 12$ by $- 1$ .

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.2.2.1.3.2

Multiply $- 1$ by $- 24$ .

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

$10 y = - y^{2} + 12 y + 24$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3

Solve for $y$ .

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Step 1.2.3.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- y^{2} + 12 y + 24 = 10 y$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.2

Move all terms containing $y$ to the left side of the equation.

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Step 1.2.3.2.1

Subtract $10 y$ from both sides of the equation.

$- y^{2} + 12 y + 24 - 10 y = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.2.2

Subtract $10 y$ from $12 y$ .

$- y^{2} + 2 y + 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

$- y^{2} + 2 y + 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3

Factor the left side of the equation.

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Step 1.2.3.3.1

Factor $- 1$ out of $- y^{2} + 2 y + 24$ .

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Step 1.2.3.3.1.1

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 2 y + 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.1.2

Factor $- 1$ out of $2 y$ .

$- (y^{2}) - (- 2 y) + 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.1.3

Rewrite $24$ as $- 1 (- 24)$ .

$- (y^{2}) - (- 2 y) - 1 \cdot - 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.1.4

Factor $- 1$ out of $- (y^{2}) - (- 2 y)$ .

$- (y^{2} - 2 y) - 1 \cdot - 24 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.1.5

Factor $- 1$ out of $- (y^{2} - 2 y) - 1 (- 24)$ .

$- (y^{2} - 2 y - 24) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

$- (y^{2} - 2 y - 24) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.2

Factor.

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Step 1.2.3.3.2.1

Factor $y^{2} - 2 y - 24$ using the AC method.

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Step 1.2.3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.2.1.2

Write the factored form using these integers.

$- ((y - 6) (y + 4)) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

$- ((y - 6) (y + 4)) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.3.2.2

Remove unnecessary parentheses.

$- (y - 6) (y + 4) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

$- (y - 6) (y + 4) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

$- (y - 6) (y + 4) = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 6 = 0$

$y + 4 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.5

Set $y - 6$ equal to $0$ and solve for $y$ .

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Step 1.2.3.5.1

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.5.2

Add $6$ to both sides of the equation.

$y = 6$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = 6$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.6

Set $y + 4$ equal to $0$ and solve for $y$ .

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Step 1.2.3.6.1

Set $y + 4$ equal to $0$ .

$y + 4 = 0$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.6.2

Subtract $4$ from both sides of the equation.

$y = - 4$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = - 4$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.2.3.7

The final solution is all the values that make $- (y - 6) (y + 4) = 0$ true.

$y = 6, - 4$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = 6, - 4$

$x = \frac{{(y - 6)}^{2}}{10}$

$y = 6, - 4$

$x = \frac{{(y - 6)}^{2}}{10}$

Step 1.3

Replace all occurrences of $y$ with $6$ in each equation.

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Step 1.3.1

Replace all occurrences of $y$ in $x = \frac{{(y - 6)}^{2}}{10}$ with $6$ .

$x = \frac{{((6) - 6)}^{2}}{10}$

$y = 6$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $\frac{{((6) - 6)}^{2}}{10}$ .

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Step 1.3.2.1.1

Simplify the numerator.

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Step 1.3.2.1.1.1

Subtract $6$ from $6$ .

$x = \frac{0^{2}}{10}$

$y = 6$

Step 1.3.2.1.1.2

Raising $0$ to any positive power yields $0$ .

$x = \frac{0}{10}$

$y = 6$

$x = \frac{0}{10}$

$y = 6$

Step 1.3.2.1.2

Divide $0$ by $10$ .

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

$x = 0$

$y = 6$

Step 1.4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 1.4.1

Replace all occurrences of $y$ in $x = \frac{{(y - 6)}^{2}}{10}$ with $- 4$ .

$x = \frac{{((- 4) - 6)}^{2}}{10}$

$y = - 4$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{{((- 4) - 6)}^{2}}{10}$ .

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Step 1.4.2.1.1

Simplify the numerator.

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Step 1.4.2.1.1.1

Subtract $6$ from $- 4$ .

$x = \frac{{(- 10)}^{2}}{10}$

$y = - 4$

Step 1.4.2.1.1.2

Raise $- 10$ to the power of $2$ .

$x = \frac{100}{10}$

$y = - 4$

$x = \frac{100}{10}$

$y = - 4$

Step 1.4.2.1.2

Divide $100$ by $10$ .

$x = 10$

$y = - 4$

$x = 10$

$y = - 4$

$x = 10$

$y = - 4$

$x = 10$

$y = - 4$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 6)$

$(10, - 4)$

$(0, 6)$

$(10, - 4)$

Step 2

Solve $y = 6 - x$ in terms of $x$ .

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Step 2.1

Rewrite the equation as $6 - x = y$ .

$6 - x = y$

Step 2.2

Subtract $6$ from both sides of the equation.

$- x = y - 6$

Step 2.3

Divide each term in $- x = y - 6$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x = y - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x = \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$x = - 1 \cdot y + \frac{- 6}{- 1}$

Step 2.3.3.1.2

Rewrite $- 1 \cdot y$ as $- y$ .

$x = - y + \frac{- 6}{- 1}$

Step 2.3.3.1.3

Divide $- 6$ by $- 1$ .

$x = - y + 6$

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 4}^{6} - y + 6 d y - \int_{- 4}^{6} \frac{{(y - 6)}^{2}}{10} d y$

Step 4

Integrate to find the area between $- 4$ and $6$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- 4}^{6} - y + 6 - (\frac{{(y - 6)}^{2}}{10}) d y$

Step 4.2

To write $- y$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\int_{- 4}^{6} - y \cdot \frac{10}{10} - \frac{{(y - 6)}^{2}}{10} + 6 d y$

Step 4.3

Simplify terms.

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Step 4.3.1

Combine $- y$ and $\frac{10}{10}$ .

$\frac{- y \cdot 10}{10} - \frac{{(y - 6)}^{2}}{10} + 6$

Step 4.3.2

Combine the numerators over the common denominator.

$\frac{- y \cdot 10 - {(y - 6)}^{2}}{10} + 6$

$\int_{- 4}^{6} \frac{- y \cdot 10 - {(y - 6)}^{2}}{10} + 6 d y$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Multiply $10$ by $- 1$ .

$\frac{- 10 y - {(y - 6)}^{2}}{10} + 6$

Step 4.4.2

Rewrite ${(y - 6)}^{2}$ as $(y - 6) (y - 6)$ .

$\frac{- 10 y - ((y - 6) (y - 6))}{10} + 6$

Step 4.4.3

Expand $(y - 6) (y - 6)$ using the FOIL Method.

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Step 4.4.3.1

Apply the distributive property.

$\frac{- 10 y - (y (y - 6) - 6 (y - 6))}{10} + 6$

Step 4.4.3.2

Apply the distributive property.

$\frac{- 10 y - (y \cdot y + y \cdot - 6 - 6 (y - 6))}{10} + 6$

Step 4.4.3.3

Apply the distributive property.

$\frac{- 10 y - (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6)}{10} + 6$

Step 4.4.4

Simplify and combine like terms.

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Step 4.4.4.1

Simplify each term.

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Step 4.4.4.1.1

Multiply $y$ by $y$ .

$\frac{- 10 y - (y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6)}{10} + 6$

Step 4.4.4.1.2

Move $- 6$ to the left of $y$ .

$\frac{- 10 y - (y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6)}{10} + 6$

Step 4.4.4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{- 10 y - (y^{2} - 6 y - 6 y + 36)}{10} + 6$

Step 4.4.4.2

Subtract $6 y$ from $- 6 y$ .

$\frac{- 10 y - (y^{2} - 12 y + 36)}{10} + 6$

Step 4.4.5

Apply the distributive property.

$\frac{- 10 y - y^{2} - (- 12 y) - 1 \cdot 36}{10} + 6$

Step 4.4.6

Simplify.

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Step 4.4.6.1

Multiply $- 12$ by $- 1$ .

$\frac{- 10 y - y^{2} + 12 y - 1 \cdot 36}{10} + 6$

Step 4.4.6.2

Multiply $- 1$ by $36$ .

$\frac{- 10 y - y^{2} + 12 y - 36}{10} + 6$

Step 4.4.7

Add $- 10 y$ and $12 y$ .

$\frac{- y^{2} + 2 y - 36}{10} + 6$

$\int_{- 4}^{6} \frac{- y^{2} + 2 y - 36}{10} + 6 d y$

Step 4.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\int_{- 4}^{6} \frac{- y^{2} + 2 y - 36}{10} + 6 \cdot \frac{10}{10} d y$

Step 4.6

Simplify terms.

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Step 4.6.1

Combine $6$ and $\frac{10}{10}$ .

$\frac{- y^{2} + 2 y - 36}{10} + \frac{6 \cdot 10}{10}$

Step 4.6.2

Combine the numerators over the common denominator.

$\frac{- y^{2} + 2 y - 36 + 6 \cdot 10}{10}$

$\int_{- 4}^{6} \frac{- y^{2} + 2 y - 36 + 6 \cdot 10}{10} d y$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $6$ by $10$ .

$\frac{- y^{2} + 2 y - 36 + 60}{10}$

Step 4.7.2

Add $- 36$ and $60$ .

$\frac{- y^{2} + 2 y + 24}{10}$

Step 4.7.3

Factor by grouping.

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Step 4.7.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 24 = - 24$ and whose sum is $b = 2$ .

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Step 4.7.3.1.1

Factor $2$ out of $2 y$ .

$\frac{- y^{2} + 2 (y) + 24}{10}$

Step 4.7.3.1.2

Rewrite $2$ as $- 4$ plus $6$

$\frac{- y^{2} + (- 4 + 6) y + 24}{10}$

Step 4.7.3.1.3

Apply the distributive property.

$\frac{- y^{2} - 4 y + 6 y + 24}{10}$

Step 4.7.3.2

Factor out the greatest common factor from each group.

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Step 4.7.3.2.1

Group the first two terms and the last two terms.

$\frac{(- y^{2} - 4 y) + 6 y + 24}{10}$

Step 4.7.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{y (- y - 4) - 6 (- y - 4)}{10}$

Step 4.7.3.3

Factor the polynomial by factoring out the greatest common factor, $- y - 4$ .

$\frac{(- y - 4) (y - 6)}{10}$

$\int_{- 4}^{6} \frac{(- y - 4) (y - 6)}{10} d y$

Step 4.8

Simplify with factoring out.

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Step 4.8.1

Factor $- 1$ out of $- y$ .

$\frac{(- (y) - 4) (y - 6)}{10}$

Step 4.8.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(- (y) - 1 (4)) (y - 6)}{10}$

Step 4.8.3

Factor $- 1$ out of $- (y) - 1 (4)$ .

$\frac{- (y + 4) (y - 6)}{10}$

Step 4.8.4

Simplify the expression.

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Step 4.8.4.1

Rewrite $- (y + 4)$ as $- 1 (y + 4)$ .

$\frac{- 1 (y + 4) (y - 6)}{10}$

Step 4.8.4.2

Move the negative in front of the fraction.

$- \frac{(y + 4) (y - 6)}{10}$

$\int_{- 4}^{6} - \frac{(y + 4) (y - 6)}{10} d y$

Step 4.9

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$- \int_{- 4}^{6} \frac{(y + 4) (y - 6)}{10} d y$

Step 4.10

Since $\frac{1}{10}$ is constant with respect to $y$ , move $\frac{1}{10}$ out of the integral.

$- (\frac{1}{10} \int_{- 4}^{6} (y + 4) (y - 6) d y)$

Step 4.11

Simplify.

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Step 4.11.1

Apply the distributive property.

$- \frac{1}{10} \int_{- 4}^{6} y (y - 6) + 4 (y - 6) d y$

Step 4.11.2

Apply the distributive property.

$- \frac{1}{10} \int_{- 4}^{6} y \cdot y + y \cdot - 6 + 4 (y - 6) d y$

Step 4.11.3

Apply the distributive property.

$- \frac{1}{10} \int_{- 4}^{6} y \cdot y + y \cdot - 6 + 4 y + 4 \cdot - 6 d y$

Step 4.11.4

Reorder $y$ and $- 6$ .

$- \frac{1}{10} \int_{- 4}^{6} y \cdot y - 6 \cdot y + 4 y + 4 \cdot - 6 d y$

Step 4.11.5

Raise $y$ to the power of $1$ .

$- \frac{1}{10} \int_{- 4}^{6} y^{1} y - 6 y + 4 y + 4 \cdot - 6 d y$

Step 4.11.6

Raise $y$ to the power of $1$ .

$- \frac{1}{10} \int_{- 4}^{6} y^{1} y^{1} - 6 y + 4 y + 4 \cdot - 6 d y$

Step 4.11.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{10} \int_{- 4}^{6} y^{1 + 1} - 6 y + 4 y + 4 \cdot - 6 d y$

Step 4.11.8

Add $1$ and $1$ .

$- \frac{1}{10} \int_{- 4}^{6} y^{2} - 6 y + 4 y + 4 \cdot - 6 d y$

Step 4.11.9

Multiply $4$ by $- 6$ .

$- \frac{1}{10} \int_{- 4}^{6} y^{2} - 6 y + 4 y - 24 d y$

Step 4.11.10

Add $- 6 y$ and $4 y$ .

$- \frac{1}{10} \int_{- 4}^{6} y^{2} - 2 y - 24 d y$

Step 4.12

Split the single integral into multiple integrals.

$- \frac{1}{10} (\int_{- 4}^{6} y^{2} d y + \int_{- 4}^{6} - 2 y d y + \int_{- 4}^{6} - 24 d y)$

Step 4.13

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$- \frac{1}{10} (\frac{1}{3} y^{3}]_{- 4}^{6} + \int_{- 4}^{6} - 2 y d y + \int_{- 4}^{6} - 24 d y)$

Step 4.14

Since $- 2$ is constant with respect to $y$ , move $- 2$ out of the integral.

$- \frac{1}{10} (\frac{1}{3} y^{3}]_{- 4}^{6} - 2 \int_{- 4}^{6} y d y + \int_{- 4}^{6} - 24 d y)$

Step 4.15

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$- \frac{1}{10} (\frac{1}{3} y^{3}]_{- 4}^{6} - 2 (\frac{1}{2} y^{2}]_{- 4}^{6}) + \int_{- 4}^{6} - 24 d y)$

Step 4.16

Simplify.

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Step 4.16.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$- \frac{1}{10} (\frac{1}{3} y^{3}]_{- 4}^{6} - 2 (\frac{y^{2}}{2}]_{- 4}^{6}) + \int_{- 4}^{6} - 24 d y)$

Step 4.16.2

Combine $\frac{1}{3}$ and $y^{3}$ .

$- \frac{1}{10} (\frac{y^{3}}{3}]_{- 4}^{6} - 2 (\frac{y^{2}}{2}]_{- 4}^{6}) + \int_{- 4}^{6} - 24 d y)$

Step 4.17

Apply the constant rule.

$- \frac{1}{10} (\frac{y^{3}}{3}]_{- 4}^{6} - 2 (\frac{y^{2}}{2}]_{- 4}^{6}) + - 24 y]_{- 4}^{6})$

Step 4.18

Simplify the answer.

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Step 4.18.1

Combine $\frac{y^{3}}{3}]_{- 4}^{6}$ and $- 24 y]_{- 4}^{6}$ .

$- \frac{1}{10} (\frac{y^{3}}{3} - 24 y]_{- 4}^{6} - 2 (\frac{y^{2}}{2}]_{- 4}^{6}))$

Step 4.18.2

Substitute and simplify.

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Step 4.18.2.1

Evaluate $\frac{y^{3}}{3} - 24 y$ at $6$ and at $- 4$ .

$- \frac{1}{10} ((\frac{6^{3}}{3} - 24 \cdot 6) - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 (\frac{y^{2}}{2}]_{- 4}^{6}))$

Step 4.18.2.2

Evaluate $\frac{y^{2}}{2}$ at $6$ and at $- 4$ .

$- \frac{1}{10} ((\frac{6^{3}}{3} - 24 \cdot 6) - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3

Simplify.

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Step 4.18.2.3.1

Raise $6$ to the power of $3$ .

$- \frac{1}{10} (\frac{216}{3} - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.2

Cancel the common factor of $216$ and $3$ .

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Step 4.18.2.3.2.1

Factor $3$ out of $216$ .

$- \frac{1}{10} (\frac{3 \cdot 72}{3} - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.2.2

Cancel the common factors.

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Step 4.18.2.3.2.2.1

Factor $3$ out of $3$ .

$- \frac{1}{10} (\frac{3 \cdot 72}{3 (1)} - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.2.2.2

Cancel the common factor.

$- \frac{1}{10} (\frac{3 \cdot 72}{3 \cdot 1} - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.2.2.3

Rewrite the expression.

$- \frac{1}{10} (\frac{72}{1} - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.2.2.4

Divide $72$ by $1$ .

$- \frac{1}{10} (72 - 24 \cdot 6 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.3

Multiply $- 24$ by $6$ .

$- \frac{1}{10} (72 - 144 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.4

Subtract $144$ from $72$ .

$- \frac{1}{10} (- 72 - (\frac{{(- 4)}^{3}}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.5

Raise $- 4$ to the power of $3$ .

$- \frac{1}{10} (- 72 - (\frac{- 64}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.6

Move the negative in front of the fraction.

$- \frac{1}{10} (- 72 - (- \frac{64}{3} - 24 \cdot - 4) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.7

Multiply $- 24$ by $- 4$ .

$- \frac{1}{10} (- 72 - (- \frac{64}{3} + 96) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.8

To write $96$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{10} (- 72 - (- \frac{64}{3} + 96 \cdot \frac{3}{3}) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.9

Combine $96$ and $\frac{3}{3}$ .

$- \frac{1}{10} (- 72 - (- \frac{64}{3} + \frac{96 \cdot 3}{3}) - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.10

Combine the numerators over the common denominator.

$- \frac{1}{10} (- 72 - \frac{- 64 + 96 \cdot 3}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.11

Simplify the numerator.

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Step 4.18.2.3.11.1

Multiply $96$ by $3$ .

$- \frac{1}{10} (- 72 - \frac{- 64 + 288}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.11.2

Add $- 64$ and $288$ .

$- \frac{1}{10} (- 72 - \frac{224}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.12

To write $- 72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{10} (- 72 \cdot \frac{3}{3} - \frac{224}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.13

Combine $- 72$ and $\frac{3}{3}$ .

$- \frac{1}{10} (\frac{- 72 \cdot 3}{3} - \frac{224}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.14

Combine the numerators over the common denominator.

$- \frac{1}{10} (\frac{- 72 \cdot 3 - 224}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.15

Simplify the numerator.

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Step 4.18.2.3.15.1

Multiply $- 72$ by $3$ .

$- \frac{1}{10} (\frac{- 216 - 224}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.15.2

Subtract $224$ from $- 216$ .

$- \frac{1}{10} (\frac{- 440}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.16

Move the negative in front of the fraction.

$- \frac{1}{10} (- \frac{440}{3} - 2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.17

Raise $6$ to the power of $2$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (\frac{36}{2} - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.18

Cancel the common factor of $36$ and $2$ .

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Step 4.18.2.3.18.1

Factor $2$ out of $36$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (\frac{2 \cdot 18}{2} - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.18.2

Cancel the common factors.

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Step 4.18.2.3.18.2.1

Factor $2$ out of $2$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (\frac{2 \cdot 18}{2 (1)} - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.18.2.2

Cancel the common factor.

$- \frac{1}{10} (- \frac{440}{3} - 2 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.18.2.3

Rewrite the expression.

$- \frac{1}{10} (- \frac{440}{3} - 2 (\frac{18}{1} - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.18.2.4

Divide $18$ by $1$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{{(- 4)}^{2}}{2}))$

Step 4.18.2.3.19

Raise $- 4$ to the power of $2$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{16}{2}))$

Step 4.18.2.3.20

Cancel the common factor of $16$ and $2$ .

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Step 4.18.2.3.20.1

Factor $2$ out of $16$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{2 \cdot 8}{2}))$

Step 4.18.2.3.20.2

Cancel the common factors.

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Step 4.18.2.3.20.2.1

Factor $2$ out of $2$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{2 \cdot 8}{2 (1)}))$

Step 4.18.2.3.20.2.2

Cancel the common factor.

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{2 \cdot 8}{2 \cdot 1}))$

Step 4.18.2.3.20.2.3

Rewrite the expression.

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - \frac{8}{1}))$

Step 4.18.2.3.20.2.4

Divide $8$ by $1$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - 1 \cdot 8))$

Step 4.18.2.3.21

Multiply $- 1$ by $8$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 (18 - 8))$

Step 4.18.2.3.22

Subtract $8$ from $18$ .

$- \frac{1}{10} (- \frac{440}{3} - 2 \cdot 10)$

Step 4.18.2.3.23

Multiply $- 2$ by $10$ .

$- \frac{1}{10} (- \frac{440}{3} - 20)$

Step 4.18.2.3.24

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{10} (- \frac{440}{3} - 20 \cdot \frac{3}{3})$

Step 4.18.2.3.25

Combine $- 20$ and $\frac{3}{3}$ .

$- \frac{1}{10} (- \frac{440}{3} + \frac{- 20 \cdot 3}{3})$

Step 4.18.2.3.26

Combine the numerators over the common denominator.

$- \frac{1}{10} \cdot \frac{- 440 - 20 \cdot 3}{3}$

Step 4.18.2.3.27

Simplify the numerator.

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Step 4.18.2.3.27.1

Multiply $- 20$ by $3$ .

$- \frac{1}{10} \cdot \frac{- 440 - 60}{3}$

Step 4.18.2.3.27.2

Subtract $60$ from $- 440$ .

$- \frac{1}{10} \cdot \frac{- 500}{3}$

Step 4.18.2.3.28

Move the negative in front of the fraction.

$- \frac{1}{10} (- \frac{500}{3})$

Step 4.18.2.3.29

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{10}) \frac{500}{3}$

Step 4.18.2.3.30

Multiply $\frac{1}{10}$ by $1$ .

$\frac{1}{10} \cdot \frac{500}{3}$

Step 4.18.2.3.31

Multiply $\frac{1}{10}$ by $\frac{500}{3}$ .

$\frac{500}{10 \cdot 3}$

Step 4.18.2.3.32

Multiply $10$ by $3$ .

$\frac{500}{30}$

Step 4.18.2.3.33

Cancel the common factor of $500$ and $30$ .

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Step 4.18.2.3.33.1

Factor $10$ out of $500$ .

$\frac{10 (50)}{30}$

Step 4.18.2.3.33.2

Cancel the common factors.

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Step 4.18.2.3.33.2.1

Factor $10$ out of $30$ .

$\frac{10 \cdot 50}{10 \cdot 3}$

Step 4.18.2.3.33.2.2

Cancel the common factor.

$\frac{10 \cdot 50}{10 \cdot 3}$

Step 4.18.2.3.33.2.3

Rewrite the expression.

$\frac{50}{3}$

Step 5