Algebra Examples

$3 x - y = 7$ $y + 4 = 2 {(x + 5)}^{2}$

Step 1

Solve for $x$ in $3 x - y = 7$ .

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Step 1.1

Add $y$ to both sides of the equation.

$3 x = 7 + y$

$y + 4 = 2 {(x + 5)}^{2}$

Step 1.2

Divide each term in $3 x = 7 + y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 7 + y$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(x + 5)}^{2}$

Step 2

Replace all occurrences of $x$ with $\frac{7}{3} + \frac{y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $y + 4 = 2 {(x + 5)}^{2}$ with $\frac{7}{3} + \frac{y}{3}$ .

$y + 4 = 2 {((\frac{7}{3} + \frac{y}{3}) + 5)}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 {((\frac{7}{3} + \frac{y}{3}) + 5)}^{2}$ .

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Step 2.2.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y + 4 = 2 {(\frac{y}{3} + \frac{7}{3} + 5 \cdot \frac{3}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.2

Combine $5$ and $\frac{3}{3}$ .

$y + 4 = 2 {(\frac{y}{3} + \frac{7}{3} + \frac{5 \cdot 3}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.3

Combine the numerators over the common denominator.

$y + 4 = 2 {(\frac{y}{3} + \frac{7 + 5 \cdot 3}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Multiply $5$ by $3$ .

$y + 4 = 2 {(\frac{y}{3} + \frac{7 + 15}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.4.2

Add $7$ and $15$ .

$y + 4 = 2 {(\frac{y}{3} + \frac{22}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 {(\frac{y}{3} + \frac{22}{3})}^{2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.5

Rewrite ${(\frac{y}{3} + \frac{22}{3})}^{2}$ as $(\frac{y}{3} + \frac{22}{3}) (\frac{y}{3} + \frac{22}{3})$ .

$y + 4 = 2 ((\frac{y}{3} + \frac{22}{3}) (\frac{y}{3} + \frac{22}{3}))$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.6

Expand $(\frac{y}{3} + \frac{22}{3}) (\frac{y}{3} + \frac{22}{3})$ using the FOIL Method.

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Step 2.2.1.6.1

Apply the distributive property.

$y + 4 = 2 (\frac{y}{3} \cdot (\frac{y}{3} + \frac{22}{3}) + \frac{22}{3} \cdot (\frac{y}{3} + \frac{22}{3}))$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.6.2

Apply the distributive property.

$y + 4 = 2 (\frac{y}{3} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot (\frac{y}{3} + \frac{22}{3}))$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.6.3

Apply the distributive property.

$y + 4 = 2 (\frac{y}{3} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y}{3} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7

Simplify and combine like terms.

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Step 2.2.1.7.1

Simplify each term.

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Step 2.2.1.7.1.1

Multiply $\frac{y}{3} \cdot \frac{y}{3}$ .

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Step 2.2.1.7.1.1.1

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

$y + 4 = 2 (\frac{y \cdot y}{3 \cdot 3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.1.2

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

$y + 4 = 2 (\frac{y \cdot y}{3 \cdot 3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.1.3

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

$y + 4 = 2 (\frac{y \cdot y}{3 \cdot 3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.1.4

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

$y + 4 = 2 (\frac{y^{1 + 1}}{3 \cdot 3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.1.5

Add $1$ and $1$ .

$y + 4 = 2 (\frac{y^{2}}{3 \cdot 3} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.1.6

Multiply $3$ by $3$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{y}{3} \cdot \frac{22}{3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.2

Multiply $\frac{y}{3} \cdot \frac{22}{3}$ .

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Step 2.2.1.7.1.2.1

Multiply $\frac{y}{3}$ by $\frac{22}{3}$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{y \cdot 22}{3 \cdot 3} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.2.2

Multiply $3$ by $3$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{y \cdot 22}{9} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{y \cdot 22}{9} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.3

Move $22$ to the left of $y$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 \cdot y}{9} + \frac{22}{3} \cdot \frac{y}{3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.4

Multiply $\frac{22}{3} \cdot \frac{y}{3}$ .

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Step 2.2.1.7.1.4.1

Multiply $\frac{22}{3}$ by $\frac{y}{3}$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{3 \cdot 3} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.4.2

Multiply $3$ by $3$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{22}{3} \cdot \frac{22}{3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.5

Multiply $\frac{22}{3} \cdot \frac{22}{3}$ .

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Step 2.2.1.7.1.5.1

Multiply $\frac{22}{3}$ by $\frac{22}{3}$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{22 \cdot 22}{3 \cdot 3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.5.2

Multiply $22$ by $22$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{484}{3 \cdot 3})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.1.5.3

Multiply $3$ by $3$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{22 y}{9} + \frac{22 y}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.7.2

Add $\frac{22 y}{9}$ and $\frac{22 y}{9}$ .

$y + 4 = 2 (\frac{y^{2}}{9} + 2 (\frac{22 y}{9}) + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + 2 (\frac{22 y}{9}) + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.8

Multiply $2 \frac{22 y}{9}$ .

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Step 2.2.1.8.1

Combine $2$ and $\frac{22 y}{9}$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{2 (22 y)}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.8.2

Multiply $22$ by $2$ .

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{44 y}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = 2 (\frac{y^{2}}{9} + \frac{44 y}{9} + \frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.9

Apply the distributive property.

$y + 4 = 2 (\frac{y^{2}}{9}) + 2 (\frac{44 y}{9}) + 2 (\frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.10

Simplify.

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Step 2.2.1.10.1

Combine $2$ and $\frac{y^{2}}{9}$ .

$y + 4 = \frac{2 y^{2}}{9} + 2 (\frac{44 y}{9}) + 2 (\frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.10.2

Multiply $2 \frac{44 y}{9}$ .

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Step 2.2.1.10.2.1

Combine $2$ and $\frac{44 y}{9}$ .

$y + 4 = \frac{2 y^{2}}{9} + \frac{2 (44 y)}{9} + 2 (\frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.10.2.2

Multiply $44$ by $2$ .

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + 2 (\frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + 2 (\frac{484}{9})$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.10.3

Multiply $2 (\frac{484}{9})$ .

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Step 2.2.1.10.3.1

Combine $2$ and $\frac{484}{9}$ .

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{2 \cdot 484}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 2.2.1.10.3.2

Multiply $2$ by $484$ .

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

$y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3

Solve for $y$ in $y + 4 = \frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9}$ .

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Step 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.

$\frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9} = y + 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2

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

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Step 3.2.1

Subtract $y$ from both sides of the equation.

$\frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{968}{9} - y = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.2

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

$\frac{2 y^{2}}{9} + \frac{88 y}{9} - y \cdot \frac{9}{9} + \frac{968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.3

Combine $- y$ and $\frac{9}{9}$ .

$\frac{2 y^{2}}{9} + \frac{88 y}{9} + \frac{- y \cdot 9}{9} + \frac{968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{2 y^{2}}{9} + \frac{88 y - y \cdot 9}{9} + \frac{968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.5

Combine the numerators over the common denominator.

$\frac{2 y^{2} + 88 y - y \cdot 9 + 968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.6

Multiply $9$ by $- 1$ .

$\frac{2 y^{2} + 88 y - 9 y + 968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.2.7

Subtract $9 y$ from $88 y$ .

$\frac{2 y^{2} + 79 y + 968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

$\frac{2 y^{2} + 79 y + 968}{9} = 4$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.3

Multiply both sides by $9$ .

$\frac{2 y^{2} + 79 y + 968}{9} \cdot 9 = 4 \cdot 9$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $9$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{2 y^{2} + 79 y + 968}{9} \cdot 9 = 4 \cdot 9$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.4.1.1.2

Rewrite the expression.

$2 y^{2} + 79 y + 968 = 4 \cdot 9$

$x = \frac{7}{3} + \frac{y}{3}$

$2 y^{2} + 79 y + 968 = 4 \cdot 9$

$x = \frac{7}{3} + \frac{y}{3}$

$2 y^{2} + 79 y + 968 = 4 \cdot 9$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $4$ by $9$ .

$2 y^{2} + 79 y + 968 = 36$

$x = \frac{7}{3} + \frac{y}{3}$

$2 y^{2} + 79 y + 968 = 36$

$x = \frac{7}{3} + \frac{y}{3}$

$2 y^{2} + 79 y + 968 = 36$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Move all terms to the left side of the equation and simplify.

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Step 3.5.1.1

Subtract $36$ from both sides of the equation.

$2 y^{2} + 79 y + 968 - 36 = 0$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.1.2

Subtract $36$ from $968$ .

$2 y^{2} + 79 y + 932 = 0$

$x = \frac{7}{3} + \frac{y}{3}$

$2 y^{2} + 79 y + 932 = 0$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.2

Use the quadratic formula to find the solutions.

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

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.3

Substitute the values $a = 2$ , $b = 79$ , and $c = 932$ into the quadratic formula and solve for $y$ .

$\frac{- 79 \pm \sqrt{79^{2} - 4 \cdot (2 \cdot 932)}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4

Simplify.

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Step 3.5.4.1

Simplify the numerator.

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Step 3.5.4.1.1

Raise $79$ to the power of $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 4 \cdot 2 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.2

Multiply $- 4 \cdot 2 \cdot 932$ .

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Step 3.5.4.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 8 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.2.2

Multiply $- 8$ by $932$ .

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.3

Subtract $7456$ from $6241$ .

$y = \frac{- 79 \pm \sqrt{- 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.4

Rewrite $- 1215$ as $- 1 (1215)$ .

$y = \frac{- 79 \pm \sqrt{- 1 \cdot 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.5

Rewrite $\sqrt{- 1 (1215)}$ as $\sqrt{- 1} \cdot \sqrt{1215}$ .

$y = \frac{- 79 \pm \sqrt{- 1} \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.6

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

$y = \frac{- 79 \pm i \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.7

Rewrite $1215$ as $9^{2} \cdot 15$ .

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Step 3.5.4.1.7.1

Factor $81$ out of $1215$ .

$y = \frac{- 79 \pm i \cdot \sqrt{81 (15)}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.7.2

Rewrite $81$ as $9^{2}$ .

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.8

Pull terms out from under the radical.

$y = \frac{- 79 \pm i \cdot (9 \sqrt{15})}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.1.9

Move $9$ to the left of $i$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.4.2

Multiply $2$ by $2$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5

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

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Step 3.5.5.1

Simplify the numerator.

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Step 3.5.5.1.1

Raise $79$ to the power of $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 4 \cdot 2 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.2

Multiply $- 4 \cdot 2 \cdot 932$ .

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Step 3.5.5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 8 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.2.2

Multiply $- 8$ by $932$ .

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.3

Subtract $7456$ from $6241$ .

$y = \frac{- 79 \pm \sqrt{- 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.4

Rewrite $- 1215$ as $- 1 (1215)$ .

$y = \frac{- 79 \pm \sqrt{- 1 \cdot 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.5

Rewrite $\sqrt{- 1 (1215)}$ as $\sqrt{- 1} \cdot \sqrt{1215}$ .

$y = \frac{- 79 \pm \sqrt{- 1} \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.6

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

$y = \frac{- 79 \pm i \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.7

Rewrite $1215$ as $9^{2} \cdot 15$ .

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Step 3.5.5.1.7.1

Factor $81$ out of $1215$ .

$y = \frac{- 79 \pm i \cdot \sqrt{81 (15)}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.7.2

Rewrite $81$ as $9^{2}$ .

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.8

Pull terms out from under the radical.

$y = \frac{- 79 \pm i \cdot (9 \sqrt{15})}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.1.9

Move $9$ to the left of $i$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.2

Multiply $2$ by $2$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.3

Change the $\pm$ to $+$ .

$y = \frac{- 79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.4

Rewrite $- 79$ as $- 1 (79)$ .

$y = \frac{- 1 \cdot 79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.5

Factor $- 1$ out of $9 i \sqrt{15}$ .

$y = \frac{- 1 \cdot 79 - (- 9 i \sqrt{15})}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.6

Factor $- 1$ out of $- 1 (79) - (- 9 i \sqrt{15})$ .

$y = \frac{- 1 (79 - 9 i \sqrt{15})}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.5.7

Move the negative in front of the fraction.

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6

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

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Step 3.5.6.1

Simplify the numerator.

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Step 3.5.6.1.1

Raise $79$ to the power of $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 4 \cdot 2 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.2

Multiply $- 4 \cdot 2 \cdot 932$ .

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Step 3.5.6.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{- 79 \pm \sqrt{6241 - 8 \cdot 932}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.2.2

Multiply $- 8$ by $932$ .

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm \sqrt{6241 - 7456}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.3

Subtract $7456$ from $6241$ .

$y = \frac{- 79 \pm \sqrt{- 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.4

Rewrite $- 1215$ as $- 1 (1215)$ .

$y = \frac{- 79 \pm \sqrt{- 1 \cdot 1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.5

Rewrite $\sqrt{- 1 (1215)}$ as $\sqrt{- 1} \cdot \sqrt{1215}$ .

$y = \frac{- 79 \pm \sqrt{- 1} \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.6

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

$y = \frac{- 79 \pm i \cdot \sqrt{1215}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.7

Rewrite $1215$ as $9^{2} \cdot 15$ .

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Step 3.5.6.1.7.1

Factor $81$ out of $1215$ .

$y = \frac{- 79 \pm i \cdot \sqrt{81 (15)}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.7.2

Rewrite $81$ as $9^{2}$ .

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm i \cdot \sqrt{9^{2} \cdot 15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.8

Pull terms out from under the radical.

$y = \frac{- 79 \pm i \cdot (9 \sqrt{15})}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.1.9

Move $9$ to the left of $i$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = \frac{- 79 \pm 9 i \sqrt{15}}{2 \cdot 2}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.2

Multiply $2$ by $2$ .

$y = \frac{- 79 \pm 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.3

Change the $\pm$ to $-$ .

$y = \frac{- 79 - 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.4

Rewrite $- 79$ as $- 1 (79)$ .

$y = \frac{- 1 \cdot 79 - 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.5

Factor $- 1$ out of $- 9 i \sqrt{15}$ .

$y = \frac{- 1 \cdot 79 - (9 i \sqrt{15})}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.6

Factor $- 1$ out of $- 1 (79) - (9 i \sqrt{15})$ .

$y = \frac{- 1 (79 + 9 i \sqrt{15})}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.6.7

Move the negative in front of the fraction.

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 3.5.7

The final answer is the combination of both solutions.

$y = - \frac{79 - 9 i \sqrt{15}}{4}, - \frac{79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}, - \frac{79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}, - \frac{79 + 9 i \sqrt{15}}{4}$

$x = \frac{7}{3} + \frac{y}{3}$

Step 4

Replace all occurrences of $y$ with $- \frac{79 - 9 i \sqrt{15}}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{7}{3} + \frac{y}{3}$ with $- \frac{79 - 9 i \sqrt{15}}{4}$ .

$x = \frac{7}{3} + \frac{- \frac{79 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{7}{3} + \frac{- \frac{79 - 9 i \sqrt{15}}{4}}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{7 - \frac{79 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{7 \cdot \frac{4}{4} - \frac{79 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.3

Combine $7$ and $\frac{4}{4}$ .

$x = \frac{\frac{7 \cdot 4}{4} - \frac{79 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{\frac{- 51 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.5

Rewrite $- 51$ as $- 1 (51)$ .

$x = \frac{\frac{- 1 \cdot 51 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.6

Factor $- 1$ out of $9 i \sqrt{15}$ .

$x = \frac{\frac{- 1 \cdot 51 - (- 9 i \sqrt{15})}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.7

Factor $- 1$ out of $- 1 (51) - (- 9 i \sqrt{15})$ .

$x = \frac{\frac{- 1 (51 - 9 i \sqrt{15})}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.8

Move the negative in front of the fraction.

$x = \frac{- \frac{51 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.9

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{51 - 9 i \sqrt{15}}{4} \cdot \frac{1}{3}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.10

Multiply $- \frac{51 - 9 i \sqrt{15}}{4} \cdot \frac{1}{3}$ .

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Step 4.2.1.10.1

Multiply $\frac{1}{3}$ by $\frac{51 - 9 i \sqrt{15}}{4}$ .

$x = - \frac{51 - 9 i \sqrt{15}}{3 \cdot 4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.10.2

Multiply $3$ by $4$ .

$x = - \frac{51 - 9 i \sqrt{15}}{12}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{51 - 9 i \sqrt{15}}{12}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11

Cancel the common factor of $51 - 9 i \sqrt{15}$ and $12$ .

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Step 4.2.1.11.1

Factor $3$ out of $51$ .

$x = - \frac{3 (17) - 9 i \sqrt{15}}{12}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11.2

Factor $3$ out of $- 9 i \sqrt{15}$ .

$x = - \frac{3 (17) + 3 (- 3 i \sqrt{15})}{12}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11.3

Factor $3$ out of $3 (17) + 3 (- 3 i \sqrt{15})$ .

$x = - \frac{3 (17 - 3 i \sqrt{15})}{12}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11.4

Cancel the common factors.

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Step 4.2.1.11.4.1

Factor $3$ out of $12$ .

$x = - \frac{3 (17 - 3 i \sqrt{15})}{3 \cdot 4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11.4.2

Cancel the common factor.

$x = - \frac{3 (17 - 3 i \sqrt{15})}{3 \cdot 4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 4.2.1.11.4.3

Rewrite the expression.

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 - 3 i \sqrt{15}}{4}$

$y = - \frac{79 - 9 i \sqrt{15}}{4}$

Step 5

Replace all occurrences of $y$ with $- \frac{79 + 9 i \sqrt{15}}{4}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{7}{3} + \frac{y}{3}$ with $- \frac{79 + 9 i \sqrt{15}}{4}$ .

$x = \frac{7}{3} + \frac{- \frac{79 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{7}{3} + \frac{- \frac{79 + 9 i \sqrt{15}}{4}}{3}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = \frac{7 - \frac{79 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{7 \cdot \frac{4}{4} - \frac{79 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.3

Combine $7$ and $\frac{4}{4}$ .

$x = \frac{\frac{7 \cdot 4}{4} - \frac{79 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x = \frac{\frac{- 51 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.5

Rewrite $- 51$ as $- 1 (51)$ .

$x = \frac{\frac{- 1 \cdot 51 - 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.6

Factor $- 1$ out of $- 9 i \sqrt{15}$ .

$x = \frac{\frac{- 1 \cdot 51 - (9 i \sqrt{15})}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.7

Factor $- 1$ out of $- 1 (51) - (9 i \sqrt{15})$ .

$x = \frac{\frac{- 1 (51 + 9 i \sqrt{15})}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.8

Move the negative in front of the fraction.

$x = \frac{- \frac{51 + 9 i \sqrt{15}}{4}}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.9

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{51 + 9 i \sqrt{15}}{4} \cdot \frac{1}{3}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.10

Multiply $- \frac{51 + 9 i \sqrt{15}}{4} \cdot \frac{1}{3}$ .

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Step 5.2.1.10.1

Multiply $\frac{1}{3}$ by $\frac{51 + 9 i \sqrt{15}}{4}$ .

$x = - \frac{51 + 9 i \sqrt{15}}{3 \cdot 4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.10.2

Multiply $3$ by $4$ .

$x = - \frac{51 + 9 i \sqrt{15}}{12}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{51 + 9 i \sqrt{15}}{12}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11

Cancel the common factor of $51 + 9 i \sqrt{15}$ and $12$ .

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Step 5.2.1.11.1

Factor $3$ out of $51$ .

$x = - \frac{3 (17) + 9 i \sqrt{15}}{12}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11.2

Factor $3$ out of $9 i \sqrt{15}$ .

$x = - \frac{3 (17) + 3 (3 i \sqrt{15})}{12}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11.3

Factor $3$ out of $3 (17) + 3 (3 i \sqrt{15})$ .

$x = - \frac{3 (17 + 3 i \sqrt{15})}{12}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11.4

Cancel the common factors.

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Step 5.2.1.11.4.1

Factor $3$ out of $12$ .

$x = - \frac{3 (17 + 3 i \sqrt{15})}{3 \cdot 4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11.4.2

Cancel the common factor.

$x = - \frac{3 (17 + 3 i \sqrt{15})}{3 \cdot 4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 5.2.1.11.4.3

Rewrite the expression.

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}$

$y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 6

List all of the solutions.

$x = - \frac{17 - 3 i \sqrt{15}}{4}, y = - \frac{79 - 9 i \sqrt{15}}{4}$

$x = - \frac{17 + 3 i \sqrt{15}}{4}, y = - \frac{79 + 9 i \sqrt{15}}{4}$

Step 7