Calculus Examples

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

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

$y = x - 4$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Simplify ${(x - 4)}^{2}$ .

Tap for more steps...

Step 1.1.2.1.1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$(x - 4) (x - 4) = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Tap for more steps...

Step 1.1.2.1.2.1

Apply the distributive property.

$x (x - 4) - 4 (x - 4) = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 (x - 4) = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 = 2 x + 7$

$y = x - 4$

$x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.3.1.2

Move $- 4$ to the left of $x$ .

$x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.3.1.3

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x - 4 x + 16 = 2 x + 7$

$y = x - 4$

$x^{2} - 4 x - 4 x + 16 = 2 x + 7$

$y = x - 4$

Step 1.1.2.1.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 8 x + 16 = 2 x + 7$

$y = x - 4$

$x^{2} - 8 x + 16 = 2 x + 7$

$y = x - 4$

$x^{2} - 8 x + 16 = 2 x + 7$

$y = x - 4$

$x^{2} - 8 x + 16 = 2 x + 7$

$y = x - 4$

$x^{2} - 8 x + 16 = 2 x + 7$

$y = x - 4$

Step 1.2

Solve for $x$ in $x^{2} - 8 x + 16 = 2 x + 7$ .

Tap for more steps...

Step 1.2.1

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

Tap for more steps...

Step 1.2.1.1

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

$x^{2} - 8 x + 16 - 2 x = 7$

$y = x - 4$

Step 1.2.1.2

Subtract $2 x$ from $- 8 x$ .

$x^{2} - 10 x + 16 = 7$

$y = x - 4$

$x^{2} - 10 x + 16 = 7$

$y = x - 4$

Step 1.2.2

Subtract $7$ from both sides of the equation.

$x^{2} - 10 x + 16 - 7 = 0$

$y = x - 4$

Step 1.2.3

Subtract $7$ from $16$ .

$x^{2} - 10 x + 9 = 0$

$y = x - 4$

Step 1.2.4

Factor $x^{2} - 10 x + 9$ using the AC method.

Tap for more steps...

Step 1.2.4.1

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

$y = x - 4$

Step 1.2.4.2

Write the factored form using these integers.

$(x - 9) (x - 1) = 0$

$y = x - 4$

$(x - 9) (x - 1) = 0$

$y = x - 4$

Step 1.2.5

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

$x - 9 = 0$

$x - 1 = 0$

$y = x - 4$

Step 1.2.6

Set $x - 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.6.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

$y = x - 4$

Step 1.2.6.2

Add $9$ to both sides of the equation.

$x = 9$

$y = x - 4$

$x = 9$

$y = x - 4$

Step 1.2.7

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = x - 4$

Step 1.2.7.2

Add $1$ to both sides of the equation.

$x = 1$

$y = x - 4$

$x = 1$

$y = x - 4$

Step 1.2.8

The final solution is all the values that make $(x - 9) (x - 1) = 0$ true.

$x = 9, 1$

$y = x - 4$

$x = 9, 1$

$y = x - 4$

Step 1.3

Replace all occurrences of $x$ with $9$ in each equation.

Tap for more steps...

Step 1.3.1

Replace all occurrences of $x$ in $y = x - 4$ with $9$ .

$y = (9) - 4$

$x = 9$

Step 1.3.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.1

Subtract $4$ from $9$ .

$y = 5$

$x = 9$

$y = 5$

$x = 9$

$y = 5$

$x = 9$

Step 1.4

Replace all occurrences of $x$ with $1$ in each equation.

Tap for more steps...

Step 1.4.1

Replace all occurrences of $x$ in $y = x - 4$ with $1$ .

$y = (1) - 4$

$x = 1$

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Subtract $4$ from $1$ .

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

Step 1.5

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 $x$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $2 x + 7 = y^{2}$ .

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

Step 2.2

Subtract $7$ from both sides of the equation.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Move the negative in front of the fraction.

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

Step 3

Solve $y = x - 4$ in terms of $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $x - 4 = y$ .

$x - 4 = y$

Step 3.2

Add $4$ to both sides of the equation.

$x = y + 4$

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.

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

Step 5

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

Tap for more steps...

Step 5.1

Combine the integrals into a single integral.

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

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2.2

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

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

Tap for more steps...

Step 5.6.1

Multiply $4$ by $2$ .

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

Step 5.6.2

Add $8$ and $7$ .

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

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

Step 5.7

Split the single integral into multiple integrals.

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

Apply the constant rule.

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

Step 5.13

Simplify the answer.

Tap for more steps...

Step 5.13.1

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

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

Step 5.13.2

Substitute and simplify.

Tap for more steps...

Step 5.13.2.1

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

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

Step 5.13.2.2

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

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

Step 5.13.2.3

Simplify.

Tap for more steps...

Step 5.13.2.3.1

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

$- \frac{1}{2} (\frac{1}{3} \cdot 125 - \frac{1}{3} {(- 3)}^{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.2

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

$- \frac{1}{2} (\frac{125}{3} - \frac{1}{3} {(- 3)}^{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.3

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

$- \frac{1}{2} (\frac{125}{3} - \frac{1}{3} \cdot - 27) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.4

Multiply $- 27$ by $- 1$ .

$- \frac{1}{2} (\frac{125}{3} + 27 (\frac{1}{3})) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.5

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

$- \frac{1}{2} (\frac{125}{3} + \frac{27}{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.6

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

Tap for more steps...

Step 5.13.2.3.6.1

Factor $3$ out of $27$ .

$- \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.6.2

Cancel the common factors.

Tap for more steps...

Step 5.13.2.3.6.2.1

Factor $3$ out of $3$ .

$- \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3 (1)}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.6.2.2

Cancel the common factor.

$- \frac{1}{2} (\frac{125}{3} + \frac{3 \cdot 9}{3 \cdot 1}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.6.2.3

Rewrite the expression.

$- \frac{1}{2} (\frac{125}{3} + \frac{9}{1}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.6.2.4

Divide $9$ by $1$ .

$- \frac{1}{2} (\frac{125}{3} + 9) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.7

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

$- \frac{1}{2} (\frac{125}{3} + 9 \cdot \frac{3}{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.8

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

$- \frac{1}{2} (\frac{125}{3} + \frac{9 \cdot 3}{3}) + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.9

Combine the numerators over the common denominator.

$- \frac{1}{2} \cdot \frac{125 + 9 \cdot 3}{3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.10

Simplify the numerator.

Tap for more steps...

Step 5.13.2.3.10.1

Multiply $9$ by $3$ .

$- \frac{1}{2} \cdot \frac{125 + 27}{3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.10.2

Add $125$ and $27$ .

$- \frac{1}{2} \cdot \frac{152}{3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.11

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

$- \frac{152}{3 \cdot 2} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.12

Multiply $3$ by $2$ .

$- \frac{152}{6} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.13

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

Tap for more steps...

Step 5.13.2.3.13.1

Factor $2$ out of $152$ .

$- \frac{2 (76)}{6} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.13.2

Cancel the common factors.

Tap for more steps...

Step 5.13.2.3.13.2.1

Factor $2$ out of $6$ .

$- \frac{2 \cdot 76}{2 \cdot 3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.13.2.2

Cancel the common factor.

$- \frac{2 \cdot 76}{2 \cdot 3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.13.2.3

Rewrite the expression.

$- \frac{76}{3} + (\frac{1}{2} \cdot 5^{2} + \frac{15}{2} \cdot 5) - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.14

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

$- \frac{76}{3} + \frac{1}{2} \cdot 25 + \frac{15}{2} \cdot 5 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.15

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

$- \frac{76}{3} + \frac{25}{2} + \frac{15}{2} \cdot 5 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.16

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

$- \frac{76}{3} + \frac{25}{2} + \frac{15 \cdot 5}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.17

Multiply $15$ by $5$ .

$- \frac{76}{3} + \frac{25}{2} + \frac{75}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.18

Combine the numerators over the common denominator.

$- \frac{76}{3} + \frac{25 + 75}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.19

Add $25$ and $75$ .

$- \frac{76}{3} + \frac{100}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.20

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

Tap for more steps...

Step 5.13.2.3.20.1

Factor $2$ out of $100$ .

$- \frac{76}{3} + \frac{2 \cdot 50}{2} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.20.2

Cancel the common factors.

Tap for more steps...

Step 5.13.2.3.20.2.1

Factor $2$ out of $2$ .

$- \frac{76}{3} + \frac{2 \cdot 50}{2 (1)} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.20.2.2

Cancel the common factor.

$- \frac{76}{3} + \frac{2 \cdot 50}{2 \cdot 1} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.20.2.3

Rewrite the expression.

$- \frac{76}{3} + \frac{50}{1} - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.20.2.4

Divide $50$ by $1$ .

$- \frac{76}{3} + 50 - (\frac{1}{2} {(- 3)}^{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.21

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

$- \frac{76}{3} + 50 - (\frac{1}{2} \cdot 9 + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.22

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

$- \frac{76}{3} + 50 - (\frac{9}{2} + \frac{15}{2} \cdot - 3)$

Step 5.13.2.3.23

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

$- \frac{76}{3} + 50 - (\frac{9}{2} + \frac{15 \cdot - 3}{2})$

Step 5.13.2.3.24

Multiply $15$ by $- 3$ .

$- \frac{76}{3} + 50 - (\frac{9}{2} + \frac{- 45}{2})$

Step 5.13.2.3.25

Move the negative in front of the fraction.

$- \frac{76}{3} + 50 - (\frac{9}{2} - \frac{45}{2})$

Step 5.13.2.3.26

Combine the numerators over the common denominator.

$- \frac{76}{3} + 50 - \frac{9 - 45}{2}$

Step 5.13.2.3.27

Subtract $45$ from $9$ .

$- \frac{76}{3} + 50 - \frac{- 36}{2}$

Step 5.13.2.3.28

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

Tap for more steps...

Step 5.13.2.3.28.1

Factor $2$ out of $- 36$ .

$- \frac{76}{3} + 50 - \frac{2 \cdot - 18}{2}$

Step 5.13.2.3.28.2

Cancel the common factors.

Tap for more steps...

Step 5.13.2.3.28.2.1

Factor $2$ out of $2$ .

$- \frac{76}{3} + 50 - \frac{2 \cdot - 18}{2 (1)}$

Step 5.13.2.3.28.2.2

Cancel the common factor.

$- \frac{76}{3} + 50 - \frac{2 \cdot - 18}{2 \cdot 1}$

Step 5.13.2.3.28.2.3

Rewrite the expression.

$- \frac{76}{3} + 50 - \frac{- 18}{1}$

Step 5.13.2.3.28.2.4

Divide $- 18$ by $1$ .

$- \frac{76}{3} + 50 - - 18$

Step 5.13.2.3.29

Multiply $- 1$ by $- 18$ .

$- \frac{76}{3} + 50 + 18$

Step 5.13.2.3.30

Add $50$ and $18$ .

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

Step 5.13.2.3.31

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

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

Step 5.13.2.3.32

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

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

Step 5.13.2.3.33

Combine the numerators over the common denominator.

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

Step 5.13.2.3.34

Simplify the numerator.

Tap for more steps...

Step 5.13.2.3.34.1

Multiply $68$ by $3$ .

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

Step 5.13.2.3.34.2

Add $- 76$ and $204$ .

$\frac{128}{3}$

Step 6