Calculus Examples

$y = x^{2}$ , $y = 2 x - 2$ , $x = 1$ , $x = 2$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

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

Step 1.2

Solve $x^{2} = 2 x - 2$ for $x$ .

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Step 1.2.1

Subtract $2 x$ from both sides of the equation.

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

Step 1.2.2

Add $2$ to both sides of the equation.

$x^{2} - 2 x + 2 = 0$

Step 1.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 2 x - 2$

Step 1.2.4

Substitute the values $a = 1$ , $b = - 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1} + y = 2 x - 2$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 1.2.5.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.5.1.4

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

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.5.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 1.2.5.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 1.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 1.2.6.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.6.1.4

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

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.6.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 1.2.6.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 1.2.6.4

Change the $\pm$ to $+$ .

$x = 1 + i$

Step 1.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 1.2.7.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 1.2.7.1.4

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

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 1.2.7.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 1.2.7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 1.2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 1.2.7.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 1.2.7.4

Change the $\pm$ to $-$ .

$x = 1 - i$

Step 1.2.8

The final answer is the combination of both solutions.

$x = 1 + i, 1 - i$

Step 1.3

Evaluate $y$ when $x = 1 + i$ .

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Step 1.3.1

Substitute $1 + i$ for $x$ .

$y = 2 (1 + i) - 2$

Step 1.3.2

Simplify $2 (1 + i) - 2$ .

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Apply the distributive property.

$y = 2 \cdot 1 + 2 i - 2$

Step 1.3.2.1.2

Multiply $2$ by $1$ .

$y = 2 + 2 i - 2$

Step 1.3.2.2

Simplify by subtracting numbers.

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Step 1.3.2.2.1

Subtract $2$ from $2$ .

$y = 0 + 2 i$

Step 1.3.2.2.2

Add $0$ and $2 i$ .

$y = 2 i$

Step 1.4

Evaluate $y$ when $x = 1 - i$ .

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Step 1.4.1

Substitute $1 - i$ for $x$ .

$y = 2 (1 - i) - 2$

Step 1.4.2

Simplify $2 (1 - i) - 2$ .

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Apply the distributive property.

$y = 2 \cdot 1 + 2 (- i) - 2$

Step 1.4.2.1.2

Multiply $2$ by $1$ .

$y = 2 + 2 (- i) - 2$

Step 1.4.2.1.3

Multiply $- 1$ by $2$ .

$y = 2 - 2 i - 2$

Step 1.4.2.2

Simplify by subtracting numbers.

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Step 1.4.2.2.1

Subtract $2$ from $2$ .

$y = 0 - 2 i$

Step 1.4.2.2.2

Subtract $2 i$ from $0$ .

$y = - 2 i$

Step 1.5

List all of the solutions.

$y = 2 i, x = 1 + i$

$y = - 2 i, x = 1 - i$

$y = 2 i, x = 1 + i$

$y = - 2 i, x = 1 - i$

Step 2

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.

$A r e a = \int_{1}^{2} x^{2} d x - \int_{1}^{2} 2 x - 2 d x$

Step 3

Integrate to find the area between $1$ and $2$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{2} x^{2} - (2 x - 2) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$x^{2} - (2 x) - - 2$

Step 3.2.2

Multiply $2$ by $- 1$ .

$x^{2} - 2 x - - 2$

Step 3.2.3

Multiply $- 1$ by $- 2$ .

$x^{2} - 2 x + 2$

$\int_{1}^{2} x^{2} - 2 x + 2 d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{1}^{2} x^{2} d x + \int_{1}^{2} - 2 x d x + \int_{1}^{2} 2 d x$

Step 3.4

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

$\frac{1}{3} x^{3}]_{1}^{2} + \int_{1}^{2} - 2 x d x + \int_{1}^{2} 2 d x$

Step 3.5

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

$\frac{1}{3} x^{3}]_{1}^{2} - 2 \int_{1}^{2} x d x + \int_{1}^{2} 2 d x$

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Apply the constant rule.

$\frac{1}{3} x^{3}]_{1}^{2} - 2 (\frac{x^{2}}{2}]_{1}^{2}) + 2 x]_{1}^{2}$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{1}{3} x^{3}]_{1}^{2}$ and $2 x]_{1}^{2}$ .

$\frac{1}{3} x^{3} + 2 x]_{1}^{2} - 2 (\frac{x^{2}}{2}]_{1}^{2})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{1}{3} x^{3} + 2 x$ at $2$ and at $1$ .

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

Step 3.9.2.2

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $1$ .

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

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

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

$\frac{1}{3} \cdot 8 + 2 \cdot 2 - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} + 2 \cdot 2 - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.3

Multiply $2$ by $2$ .

$\frac{8}{3} + 4 - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} + 4 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.5

Combine $4$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{4 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.6

Combine the numerators over the common denominator.

$\frac{8 + 4 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.7

Simplify the numerator.

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Step 3.9.2.3.7.1

Multiply $4$ by $3$ .

$\frac{8 + 12}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.7.2

Add $8$ and $12$ .

$\frac{20}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.8

One to any power is one.

$\frac{20}{3} - (\frac{1}{3} \cdot 1 + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.9

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

$\frac{20}{3} - (\frac{1}{3} + 2 \cdot 1) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.10

Multiply $2$ by $1$ .

$\frac{20}{3} - (\frac{1}{3} + 2) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.11

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{20}{3} - (\frac{1}{3} + 2 \cdot \frac{3}{3}) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.12

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

$\frac{20}{3} - (\frac{1}{3} + \frac{2 \cdot 3}{3}) - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.13

Combine the numerators over the common denominator.

$\frac{20}{3} - \frac{1 + 2 \cdot 3}{3} - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.14

Simplify the numerator.

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Step 3.9.2.3.14.1

Multiply $2$ by $3$ .

$\frac{20}{3} - \frac{1 + 6}{3} - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.14.2

Add $1$ and $6$ .

$\frac{20}{3} - \frac{7}{3} - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.15

Combine the numerators over the common denominator.

$\frac{20 - 7}{3} - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.16

Subtract $7$ from $20$ .

$\frac{13}{3} - 2 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2})$

Step 3.9.2.3.17

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

$\frac{13}{3} - 2 (\frac{4}{2} - \frac{1^{2}}{2})$

Step 3.9.2.3.18

Cancel the common factor of $4$ and $2$ .

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Step 3.9.2.3.18.1

Factor $2$ out of $4$ .

$\frac{13}{3} - 2 (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2})$

Step 3.9.2.3.18.2

Cancel the common factors.

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Step 3.9.2.3.18.2.1

Factor $2$ out of $2$ .

$\frac{13}{3} - 2 (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2})$

Step 3.9.2.3.18.2.2

Cancel the common factor.

$\frac{13}{3} - 2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2})$

Step 3.9.2.3.18.2.3

Rewrite the expression.

$\frac{13}{3} - 2 (\frac{2}{1} - \frac{1^{2}}{2})$

Step 3.9.2.3.18.2.4

Divide $2$ by $1$ .

$\frac{13}{3} - 2 (2 - \frac{1^{2}}{2})$

Step 3.9.2.3.19

One to any power is one.

$\frac{13}{3} - 2 (2 - \frac{1}{2})$

Step 3.9.2.3.20

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

$\frac{13}{3} - 2 (2 \cdot \frac{2}{2} - \frac{1}{2})$

Step 3.9.2.3.21

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

$\frac{13}{3} - 2 (\frac{2 \cdot 2}{2} - \frac{1}{2})$

Step 3.9.2.3.22

Combine the numerators over the common denominator.

$\frac{13}{3} - 2 \frac{2 \cdot 2 - 1}{2}$

Step 3.9.2.3.23

Simplify the numerator.

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Step 3.9.2.3.23.1

Multiply $2$ by $2$ .

$\frac{13}{3} - 2 \frac{4 - 1}{2}$

Step 3.9.2.3.23.2

Subtract $1$ from $4$ .

$\frac{13}{3} - 2 (\frac{3}{2})$

Step 3.9.2.3.24

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

$\frac{13}{3} + \frac{- 2 \cdot 3}{2}$

Step 3.9.2.3.25

Multiply $- 2$ by $3$ .

$\frac{13}{3} + \frac{- 6}{2}$

Step 3.9.2.3.26

Cancel the common factor of $- 6$ and $2$ .

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Step 3.9.2.3.26.1

Factor $2$ out of $- 6$ .

$\frac{13}{3} + \frac{2 \cdot - 3}{2}$

Step 3.9.2.3.26.2

Cancel the common factors.

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Step 3.9.2.3.26.2.1

Factor $2$ out of $2$ .

$\frac{13}{3} + \frac{2 \cdot - 3}{2 (1)}$

Step 3.9.2.3.26.2.2

Cancel the common factor.

$\frac{13}{3} + \frac{2 \cdot - 3}{2 \cdot 1}$

Step 3.9.2.3.26.2.3

Rewrite the expression.

$\frac{13}{3} + \frac{- 3}{1}$

Step 3.9.2.3.26.2.4

Divide $- 3$ by $1$ .

$\frac{13}{3} - 3$

Step 3.9.2.3.27

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

$\frac{13}{3} - 3 \cdot \frac{3}{3}$

Step 3.9.2.3.28

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

$\frac{13}{3} + \frac{- 3 \cdot 3}{3}$

Step 3.9.2.3.29

Combine the numerators over the common denominator.

$\frac{13 - 3 \cdot 3}{3}$

Step 3.9.2.3.30

Simplify the numerator.

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Step 3.9.2.3.30.1

Multiply $- 3$ by $3$ .

$\frac{13 - 9}{3}$

Step 3.9.2.3.30.2

Subtract $9$ from $13$ .

$\frac{4}{3}$

Step 4