Calculus Examples

$y^{2} - 2 x = 7$ , $x - y = 4$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Add $y$ to both sides of the equation.

$x = 4 + y$

$y^{2} - 2 x = 7$

Step 1.2

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

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Step 1.2.1

Replace all occurrences of $x$ in $y^{2} - 2 x = 7$ with $4 + y$ .

$y^{2} - 2 (4 + y) = 7$

$x = 4 + y$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Apply the distributive property.

$y^{2} - 2 \cdot 4 - 2 y = 7$

$x = 4 + y$

Step 1.2.2.1.2

Multiply $- 2$ by $4$ .

$y^{2} - 8 - 2 y = 7$

$x = 4 + y$

$y^{2} - 8 - 2 y = 7$

$x = 4 + y$

$y^{2} - 8 - 2 y = 7$

$x = 4 + y$

$y^{2} - 8 - 2 y = 7$

$x = 4 + y$

Step 1.3

Solve for $y$ in $y^{2} - 8 - 2 y = 7$ .

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Step 1.3.1

Subtract $7$ from both sides of the equation.

$y^{2} - 8 - 2 y - 7 = 0$

$x = 4 + y$

Step 1.3.2

Subtract $7$ from $- 8$ .

$y^{2} - 2 y - 15 = 0$

$x = 4 + y$

Step 1.3.3

Factor $y^{2} - 2 y - 15$ using the AC method.

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Step 1.3.3.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

$x = 4 + y$

Step 1.3.3.2

Write the factored form using these integers.

$(y - 5) (y + 3) = 0$

$x = 4 + y$

$(y - 5) (y + 3) = 0$

$x = 4 + y$

Step 1.3.4

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

$y - 5 = 0$

$y + 3 = 0$

$x = 4 + y$

Step 1.3.5

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 1.3.5.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

$x = 4 + y$

Step 1.3.5.2

Add $5$ to both sides of the equation.

$y = 5$

$x = 4 + y$

$y = 5$

$x = 4 + y$

Step 1.3.6

Set $y + 3$ equal to $0$ and solve for $y$ .

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Step 1.3.6.1

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

$x = 4 + y$

Step 1.3.6.2

Subtract $3$ from both sides of the equation.

$y = - 3$

$x = 4 + y$

$y = - 3$

$x = 4 + y$

Step 1.3.7

The final solution is all the values that make $(y - 5) (y + 3) = 0$ true.

$y = 5, - 3$

$x = 4 + y$

$y = 5, - 3$

$x = 4 + y$

Step 1.4

Replace all occurrences of $y$ with $5$ in each equation.

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Step 1.4.1

Replace all occurrences of $y$ in $x = 4 + y$ with $5$ .

$x = 4 + 5$

$y = 5$

Step 1.4.2

Simplify $x = 4 + 5$ .

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Step 1.4.2.1

Simplify the left side.

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Step 1.4.2.1.1

Remove parentheses.

$x = 4 + 5$

$y = 5$

$x = 4 + 5$

$y = 5$

Step 1.4.2.2

Simplify the right side.

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Step 1.4.2.2.1

Add $4$ and $5$ .

$x = 9$

$y = 5$

$x = 9$

$y = 5$

$x = 9$

$y = 5$

$x = 9$

$y = 5$

Step 1.5

Replace all occurrences of $y$ with $- 3$ in each equation.

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Step 1.5.1

Replace all occurrences of $y$ in $x = 4 + y$ with $- 3$ .

$x = 4 - 3$

$y = - 3$

Step 1.5.2

Simplify $x = 4 - 3$ .

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Remove parentheses.

$x = 4 - 3$

$y = - 3$

$x = 4 - 3$

$y = - 3$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Subtract $3$ from $4$ .

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

Step 1.6

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

$(9, 5)$

$(1, - 3)$

$(9, 5)$

$(1, - 3)$

Step 2

Solve $y^{2} - 2 x = 7$ in terms of $x$ .

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Step 2.1

Subtract $y^{2}$ from both sides of the equation.

$- 2 x = 7 - y^{2}$

Step 2.2

Divide each term in $- 2 x = 7 - y^{2}$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x = 7 - y^{2}$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{7}{- 2} + \frac{- y^{2}}{- 2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{7}{- 2} + \frac{- y^{2}}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{- 2} + \frac{- y^{2}}{- 2}$

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

Move the negative in front of the fraction.

$x = - \frac{7}{2} + \frac{- y^{2}}{- 2}$

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{7}{2} + \frac{y^{2}}{2}$

Step 3

Add $y$ to both sides of the equation.

$x = 4 + y$

Step 4

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.

$A r e a = \int_{- 3}^{5} 4 + y d y - \int_{- 3}^{5} - \frac{7}{2} + \frac{y^{2}}{2} d y$

Step 5

Integrate to find the area between $- 3$ and $5$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{- 3}^{5} 4 + y - (- \frac{7}{2} + \frac{y^{2}}{2}) d y$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$4 + y - - \frac{7}{2} - \frac{y^{2}}{2}$

Step 5.2.2

Multiply $- - \frac{7}{2}$ .

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Step 5.2.2.1

Multiply $- 1$ by $- 1$ .

$4 + y + 1 (\frac{7}{2}) - \frac{y^{2}}{2}$

Step 5.2.2.2

Multiply $\frac{7}{2}$ by $1$ .

$4 + y + \frac{7}{2} - \frac{y^{2}}{2}$

$\int_{- 3}^{5} 4 + y + \frac{7}{2} - \frac{y^{2}}{2} d y$

Step 5.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int_{- 3}^{5} y + 4 \cdot \frac{2}{2} + \frac{7}{2} - \frac{y^{2}}{2} d y$

Step 5.4

Combine $4$ and $\frac{2}{2}$ .

$\int_{- 3}^{5} y + \frac{4 \cdot 2}{2} + \frac{7}{2} - \frac{y^{2}}{2} d y$

Step 5.5

Combine the numerators over the common denominator.

$\int_{- 3}^{5} y + \frac{4 \cdot 2 + 7}{2} - \frac{y^{2}}{2} d y$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $4$ by $2$ .

$y + \frac{8 + 7}{2} - \frac{y^{2}}{2}$

Step 5.6.2

Add $8$ and $7$ .

$y + \frac{15}{2} - \frac{y^{2}}{2}$

$\int_{- 3}^{5} y + \frac{15}{2} - \frac{y^{2}}{2} d y$

Step 5.7

Split the single integral into multiple integrals.

$\int_{- 3}^{5} y d y + \int_{- 3}^{5} \frac{15}{2} d y + \int_{- 3}^{5} - \frac{y^{2}}{2} d y$

Step 5.8

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

$\frac{1}{2} y^{2}]_{- 3}^{5} + \int_{- 3}^{5} \frac{15}{2} d y + \int_{- 3}^{5} - \frac{y^{2}}{2} d y$

Step 5.9

Apply the constant rule.

$\frac{1}{2} y^{2}]_{- 3}^{5} + \frac{15}{2} y]_{- 3}^{5} + \int_{- 3}^{5} - \frac{y^{2}}{2} d y$

Step 5.10

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

$\frac{1}{2} y^{2}]_{- 3}^{5} + \frac{15}{2} y]_{- 3}^{5} - \int_{- 3}^{5} \frac{y^{2}}{2} d y$

Step 5.11

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} y^{2}]_{- 3}^{5} + \frac{15}{2} y]_{- 3}^{5} - (\frac{1}{2} \int_{- 3}^{5} y^{2} d y)$

Step 5.12

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

$\frac{1}{2} y^{2}]_{- 3}^{5} + \frac{15}{2} y]_{- 3}^{5} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 3}^{5}$

Step 5.13

Simplify the answer.

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Step 5.13.1

Combine $\frac{1}{2} y^{2}]_{- 3}^{5}$ and $\frac{15}{2} y]_{- 3}^{5}$ .

$\frac{1}{2} y^{2} + \frac{15}{2} y]_{- 3}^{5} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 3}^{5}$

Step 5.13.2

Substitute and simplify.

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Step 5.13.2.1

Evaluate $\frac{1}{2} y^{2} + \frac{15}{2} y$ at $5$ and at $- 3$ .

$(\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} \frac{1}{3} y^{3}]_{- 3}^{5}$

Step 5.13.2.2

Evaluate $\frac{1}{3} y^{3}$ at $5$ and at $- 3$ .

$(\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3

Simplify.

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Step 5.13.2.3.1

Raise $5$ to the power of $2$ .

$\frac{1}{2} \cdot 25 + \frac{15}{2} \cdot 5 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.2

Combine $\frac{1}{2}$ and $25$ .

$\frac{25}{2} + \frac{15}{2} \cdot 5 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.3

Combine $\frac{15}{2}$ and $5$ .

$\frac{25}{2} + \frac{15 \cdot 5}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.4

Multiply $15$ by $5$ .

$\frac{25}{2} + \frac{75}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.5

Combine the numerators over the common denominator.

$\frac{25 + 75}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.6

Add $25$ and $75$ .

$\frac{100}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.7

Cancel the common factor of $100$ and $2$ .

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Step 5.13.2.3.7.1

Factor $2$ out of $100$ .

$\frac{2 \cdot 50}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.7.2

Cancel the common factors.

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Step 5.13.2.3.7.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 50}{2 (1)} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.7.2.2

Cancel the common factor.

$\frac{2 \cdot 50}{2 \cdot 1} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.7.2.3

Rewrite the expression.

$\frac{50}{1} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.7.2.4

Divide $50$ by $1$ .

$50 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.8

Raise $- 3$ to the power of $2$ .

$50 - (\frac{1}{2} \cdot 9 + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.9

Combine $\frac{1}{2}$ and $9$ .

$50 - (\frac{9}{2} + \frac{15}{2} \cdot - 3) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.10

Combine $\frac{15}{2}$ and $- 3$ .

$50 - (\frac{9}{2} + \frac{15 \cdot - 3}{2}) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.11

Multiply $15$ by $- 3$ .

$50 - (\frac{9}{2} + \frac{- 45}{2}) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.12

Move the negative in front of the fraction.

$50 - (\frac{9}{2} - \frac{45}{2}) - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.13

Combine the numerators over the common denominator.

$50 - \frac{9 - 45}{2} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.14

Subtract $45$ from $9$ .

$50 - \frac{- 36}{2} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.15

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

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Step 5.13.2.3.15.1

Factor $2$ out of $- 36$ .

$50 - \frac{2 \cdot - 18}{2} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.15.2

Cancel the common factors.

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Step 5.13.2.3.15.2.1

Factor $2$ out of $2$ .

$50 - \frac{2 \cdot - 18}{2 (1)} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.15.2.2

Cancel the common factor.

$50 - \frac{2 \cdot - 18}{2 \cdot 1} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.15.2.3

Rewrite the expression.

$50 - \frac{- 18}{1} - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.15.2.4

Divide $- 18$ by $1$ .

$50 - - 18 - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.16

Multiply $- 1$ by $- 18$ .

$50 + 18 - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.17

Add $50$ and $18$ .

$68 - \frac{1}{2} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.18

Raise $5$ to the power of $3$ .

$68 - \frac{1}{2} (\frac{1}{3} \cdot 125 - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.19

Combine $\frac{1}{3}$ and $125$ .

$68 - \frac{1}{2} (\frac{125}{3} - \frac{1}{3} {(- 3)}^{3})$

Step 5.13.2.3.20

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

$68 - \frac{1}{2} (\frac{125}{3} - \frac{1}{3} \cdot - 27)$

Step 5.13.2.3.21

Multiply $- 27$ by $- 1$ .

$68 - \frac{1}{2} (\frac{125}{3} + 27 (\frac{1}{3}))$

Step 5.13.2.3.22

Combine $27$ and $\frac{1}{3}$ .

$68 - \frac{1}{2} (\frac{125}{3} + \frac{27}{3})$

Step 5.13.2.3.23

Cancel the common factor of $27$ and $3$ .

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Step 5.13.2.3.23.1

Factor $3$ out of $27$ .

$68 - \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3})$

Step 5.13.2.3.23.2

Cancel the common factors.

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Step 5.13.2.3.23.2.1

Factor $3$ out of $3$ .

$68 - \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3 (1)})$

Step 5.13.2.3.23.2.2

Cancel the common factor.

$68 - \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3 \cdot 1})$

Step 5.13.2.3.23.2.3

Rewrite the expression.

$68 - \frac{1}{2} (\frac{125}{3} + \frac{9}{1})$

Step 5.13.2.3.23.2.4

Divide $9$ by $1$ .

$68 - \frac{1}{2} (\frac{125}{3} + 9)$

Step 5.13.2.3.24

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$68 - \frac{1}{2} (\frac{125}{3} + 9 \cdot \frac{3}{3})$

Step 5.13.2.3.25

Combine $9$ and $\frac{3}{3}$ .

$68 - \frac{1}{2} (\frac{125}{3} + \frac{9 \cdot 3}{3})$

Step 5.13.2.3.26

Combine the numerators over the common denominator.

$68 - \frac{1}{2} \cdot \frac{125 + 9 \cdot 3}{3}$

Step 5.13.2.3.27

Simplify the numerator.

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Step 5.13.2.3.27.1

Multiply $9$ by $3$ .

$68 - \frac{1}{2} \cdot \frac{125 + 27}{3}$

Step 5.13.2.3.27.2

Add $125$ and $27$ .

$68 - \frac{1}{2} \cdot \frac{152}{3}$

Step 5.13.2.3.28

Multiply $\frac{152}{3}$ by $\frac{1}{2}$ .

$68 - \frac{152}{3 \cdot 2}$

Step 5.13.2.3.29

Multiply $3$ by $2$ .

$68 - \frac{152}{6}$

Step 5.13.2.3.30

Cancel the common factor of $152$ and $6$ .

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Step 5.13.2.3.30.1

Factor $2$ out of $152$ .

$68 - \frac{2 (76)}{6}$

Step 5.13.2.3.30.2

Cancel the common factors.

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Step 5.13.2.3.30.2.1

Factor $2$ out of $6$ .

$68 - \frac{2 \cdot 76}{2 \cdot 3}$

Step 5.13.2.3.30.2.2

Cancel the common factor.

$68 - \frac{2 \cdot 76}{2 \cdot 3}$

Step 5.13.2.3.30.2.3

Rewrite the expression.

$68 - \frac{76}{3}$

Step 5.13.2.3.31

To write $68$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$68 \cdot \frac{3}{3} - \frac{76}{3}$

Step 5.13.2.3.32

Combine $68$ and $\frac{3}{3}$ .

$\frac{68 \cdot 3}{3} - \frac{76}{3}$

Step 5.13.2.3.33

Combine the numerators over the common denominator.

$\frac{68 \cdot 3 - 76}{3}$

Step 5.13.2.3.34

Simplify the numerator.

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Step 5.13.2.3.34.1

Multiply $68$ by $3$ .

$\frac{204 - 76}{3}$

Step 5.13.2.3.34.2

Subtract $76$ from $204$ .

$\frac{128}{3}$

Step 6