Finite Math Examples

$4 x^{2} + y^{2} = 1$ , $x^{2} - 3 y^{2} = 14$

Step 1

Solve for $x$ in $x^{2} - 3 y^{2} = 14$ .

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Step 1.1

Add $3 y^{2}$ to both sides of the equation.

$x^{2} = 14 + 3 y^{2}$

$4 x^{2} + y^{2} = 1$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

Step 1.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

Step 1.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

Step 1.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{14 + 3 y^{2}}$

$x = - \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$x = - \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$x = - \sqrt{14 + 3 y^{2}}$

$4 x^{2} + y^{2} = 1$

Step 2

Solve the system $x = \sqrt{14 + 3 y^{2}}, 4 x^{2} + y^{2} = 1$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{14 + 3 y^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $4 x^{2} + y^{2} = 1$ with $\sqrt{14 + 3 y^{2}}$ .

$4 {(\sqrt{14 + 3 y^{2}})}^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $4 {(\sqrt{14 + 3 y^{2}})}^{2} + y^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{14 + 3 y^{2}}}^{2}$ as $14 + 3 y^{2}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14 + 3 y^{2}}$ as ${(14 + 3 y^{2})}^{\frac{1}{2}}$ .

$4 {({(14 + 3 y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(14 + 3 y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.1.3

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

$4 {(14 + 3 y^{2})}^{\frac{2}{2}} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

$4 {(14 + 3 y^{2})}^{\frac{2}{2}} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.1.5

Simplify.

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.2

Apply the distributive property.

$4 \cdot 14 + 4 (3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.3

Multiply $4$ by $14$ .

$56 + 4 (3 y^{2}) + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.1.4

Multiply $3$ by $4$ .

$56 + 12 y^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$56 + 12 y^{2} + y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.1.2.1.2

Add $12 y^{2}$ and $y^{2}$ .

$56 + 13 y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2

Solve for $y$ in $56 + 13 y^{2} = 1$ .

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Step 2.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.1.1

Subtract $56$ from both sides of the equation.

$13 y^{2} = 1 - 56$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.1.2

Subtract $56$ from $1$ .

$13 y^{2} = - 55$

$x = \sqrt{14 + 3 y^{2}}$

$13 y^{2} = - 55$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.2

Divide each term in $13 y^{2} = - 55$ by $13$ and simplify.

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Step 2.2.2.1

Divide each term in $13 y^{2} = - 55$ by $13$ .

$\frac{13 y^{2}}{13} = \frac{- 55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $13$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{13 y^{2}}{13} = \frac{- 55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y^{2} = \frac{- 55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y^{2} = \frac{- 55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y^{2} = - \frac{55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y^{2} = - \frac{55}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{55}{13}}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4

Simplify $\pm \sqrt{- \frac{55}{13}}$ .

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Step 2.2.4.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} (\frac{55}{13})}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.2

Pull terms out from under the radical.

$y = \pm i \sqrt{\frac{55}{13}}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.3

Rewrite $\sqrt{\frac{55}{13}}$ as $\frac{\sqrt{55}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55}}{\sqrt{13}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.4

Multiply $\frac{\sqrt{55}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5

Combine and simplify the denominator.

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Step 2.2.4.5.1

Multiply $\frac{\sqrt{55}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.2

Raise $\sqrt{13}$ to the power of $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.3

Raise $\sqrt{13}$ to the power of $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.4

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

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{\sqrt{13}}^{1 + 1}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.5

Add $1$ and $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{\sqrt{13}}^{2}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6.3

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

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{2}{2}}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.5.6.4.1

Cancel the common factor.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{2}{2}}})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6.4.2

Rewrite the expression.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.5.6.5

Evaluate the exponent.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.6

Simplify the numerator.

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Step 2.2.4.6.1

Combine using the product rule for radicals.

$y = \pm i (\frac{\sqrt{55 \cdot 13}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.6.2

Multiply $55$ by $13$ .

$y = \pm i (\frac{\sqrt{715}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{715}}{13})$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.4.7

Combine $i$ and $\frac{\sqrt{715}}{13}$ .

$y = \pm \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y = \pm \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 y^{2}}$

Step 2.3

Replace all occurrences of $y$ with $\frac{i \sqrt{715}}{13}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{14 + 3 y^{2}}$ with $\frac{i \sqrt{715}}{13}$ .

$x = \sqrt{14 + 3 {(\frac{i \sqrt{715}}{13})}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{14 + 3 {(\frac{i \sqrt{715}}{13})}^{2}}$ .

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Step 2.3.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.2.1.1.1

Apply the product rule to $\frac{i \sqrt{715}}{13}$ .

$x = \sqrt{14 + 3 (\frac{{(i \sqrt{715})}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.1.2

Apply the product rule to $i \sqrt{715}$ .

$x = \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2

Simplify the numerator.

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Step 2.3.2.1.2.1

Rewrite $i^{2}$ as $- 1$ .

$x = \sqrt{14 + 3 (\frac{- 1 {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2

Rewrite ${\sqrt{715}}^{2}$ as $715$ .

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Step 2.3.2.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{715}$ as $715^{\frac{1}{2}}$ .

$x = \sqrt{14 + 3 (\frac{- 1 {(715^{\frac{1}{2}})}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{1}{2} \cdot 2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2.3

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

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.2.4.1

Cancel the common factor.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2.4.2

Rewrite the expression.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.2.2.5

Evaluate the exponent.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3

Reduce the expression by cancelling the common factors.

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Step 2.3.2.1.3.1

Raise $13$ to the power of $2$ .

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.2

Multiply $- 1$ by $715$ .

$x = \sqrt{14 + 3 (\frac{- 715}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.3

Cancel the common factor of $- 715$ and $169$ .

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Step 2.3.2.1.3.3.1

Factor $13$ out of $- 715$ .

$x = \sqrt{14 + 3 (\frac{13 (- 55)}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.3.2

Cancel the common factors.

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Step 2.3.2.1.3.3.2.1

Factor $13$ out of $169$ .

$x = \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.3.2.2

Cancel the common factor.

$x = \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.3.2.3

Rewrite the expression.

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.3.4

Move the negative in front of the fraction.

$x = \sqrt{14 + 3 (- \frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (- \frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.4

Multiply $3 (- \frac{55}{13})$ .

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Step 2.3.2.1.4.1

Multiply $- 1$ by $3$ .

$x = \sqrt{14 - 3 (\frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.4.2

Combine $- 3$ and $\frac{55}{13}$ .

$x = \sqrt{14 + \frac{- 3 \cdot 55}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.4.3

Multiply $- 3$ by $55$ .

$x = \sqrt{14 + \frac{- 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + \frac{- 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.5

Move the negative in front of the fraction.

$x = \sqrt{14 - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.6

To write $14$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = \sqrt{14 \cdot \frac{13}{13} - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.7

Combine $14$ and $\frac{13}{13}$ .

$x = \sqrt{\frac{14 \cdot 13}{13} - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.8

Combine the numerators over the common denominator.

$x = \sqrt{\frac{14 \cdot 13 - 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.9

Simplify the numerator.

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Step 2.3.2.1.9.1

Multiply $14$ by $13$ .

$x = \sqrt{\frac{182 - 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.9.2

Subtract $165$ from $182$ .

$x = \sqrt{\frac{17}{13}}$

$y = \frac{i \sqrt{715}}{13}$

$x = \sqrt{\frac{17}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.10

Rewrite $\sqrt{\frac{17}{13}}$ as $\frac{\sqrt{17}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17}}{\sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.11

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12

Combine and simplify the denominator.

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Step 2.3.2.1.12.1

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.2

Raise $\sqrt{13}$ to the power of $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.3

Raise $\sqrt{13}$ to the power of $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.4

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

$x = \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.5

Add $1$ and $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.3.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6.3

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

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.12.6.4.1

Cancel the common factor.

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6.4.2

Rewrite the expression.

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.12.6.5

Evaluate the exponent.

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.13

Simplify the numerator.

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Step 2.3.2.1.13.1

Combine using the product rule for radicals.

$x = \frac{\sqrt{17 \cdot 13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.3.2.1.13.2

Multiply $17$ by $13$ .

$x = \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 2.4

Replace all occurrences of $y$ with $- \frac{i \sqrt{715}}{13}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{14 + 3 y^{2}}$ with $- \frac{i \sqrt{715}}{13}$ .

$x = \sqrt{14 + 3 {(- \frac{i \sqrt{715}}{13})}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{14 + 3 {(- \frac{i \sqrt{715}}{13})}^{2}}$ .

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Step 2.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.2.1.1.1

Apply the product rule to $- \frac{i \sqrt{715}}{13}$ .

$x = \sqrt{14 + 3 ({(- 1)}^{2} {(\frac{i \sqrt{715}}{13})}^{2})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.1.2

Apply the product rule to $\frac{i \sqrt{715}}{13}$ .

$x = \sqrt{14 + 3 ({(- 1)}^{2} (\frac{{(i \sqrt{715})}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.1.3

Apply the product rule to $i \sqrt{715}$ .

$x = \sqrt{14 + 3 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.2

Simplify the expression.

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Step 2.4.2.1.2.1

Raise $- 1$ to the power of $2$ .

$x = \sqrt{14 + 3 (1 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.2.2

Multiply $\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}$ by $1$ .

$x = \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3

Simplify the numerator.

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Step 2.4.2.1.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = \sqrt{14 + 3 (\frac{- 1 {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2

Rewrite ${\sqrt{715}}^{2}$ as $715$ .

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Step 2.4.2.1.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{715}$ as $715^{\frac{1}{2}}$ .

$x = \sqrt{14 + 3 (\frac{- 1 {(715^{\frac{1}{2}})}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{1}{2} \cdot 2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2.3

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

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.3.2.4.1

Cancel the common factor.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2.4.2

Rewrite the expression.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.3.2.5

Evaluate the exponent.

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 2.4.2.1.4.1

Raise $13$ to the power of $2$ .

$x = \sqrt{14 + 3 (\frac{- 1 \cdot 715}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.2

Multiply $- 1$ by $715$ .

$x = \sqrt{14 + 3 (\frac{- 715}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.3

Cancel the common factor of $- 715$ and $169$ .

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Step 2.4.2.1.4.3.1

Factor $13$ out of $- 715$ .

$x = \sqrt{14 + 3 (\frac{13 (- 55)}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.3.2

Cancel the common factors.

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Step 2.4.2.1.4.3.2.1

Factor $13$ out of $169$ .

$x = \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.3.2.2

Cancel the common factor.

$x = \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.3.2.3

Rewrite the expression.

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.4.4

Move the negative in front of the fraction.

$x = \sqrt{14 + 3 (- \frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + 3 (- \frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.5

Multiply $3 (- \frac{55}{13})$ .

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Step 2.4.2.1.5.1

Multiply $- 1$ by $3$ .

$x = \sqrt{14 - 3 (\frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.5.2

Combine $- 3$ and $\frac{55}{13}$ .

$x = \sqrt{14 + \frac{- 3 \cdot 55}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.5.3

Multiply $- 3$ by $55$ .

$x = \sqrt{14 + \frac{- 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{14 + \frac{- 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.6

Move the negative in front of the fraction.

$x = \sqrt{14 - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.7

To write $14$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = \sqrt{14 \cdot \frac{13}{13} - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.8

Combine $14$ and $\frac{13}{13}$ .

$x = \sqrt{\frac{14 \cdot 13}{13} - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.9

Combine the numerators over the common denominator.

$x = \sqrt{\frac{14 \cdot 13 - 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.10

Simplify the numerator.

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Step 2.4.2.1.10.1

Multiply $14$ by $13$ .

$x = \sqrt{\frac{182 - 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.10.2

Subtract $165$ from $182$ .

$x = \sqrt{\frac{17}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \sqrt{\frac{17}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.11

Rewrite $\sqrt{\frac{17}{13}}$ as $\frac{\sqrt{17}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17}}{\sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.12

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13

Combine and simplify the denominator.

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Step 2.4.2.1.13.1

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.2

Raise $\sqrt{13}$ to the power of $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.3

Raise $\sqrt{13}$ to the power of $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.4

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

$x = \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.5

Add $1$ and $1$ .

$x = \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.4.2.1.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6.3

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

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.13.6.4.1

Cancel the common factor.

$x = \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6.4.2

Rewrite the expression.

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.13.6.5

Evaluate the exponent.

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.14

Simplify the numerator.

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Step 2.4.2.1.14.1

Combine using the product rule for radicals.

$x = \frac{\sqrt{17 \cdot 13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 2.4.2.1.14.2

Multiply $17$ by $13$ .

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3

Solve the system $x = - \sqrt{14 + 3 y^{2}}, 4 x^{2} + y^{2} = 1$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{14 + 3 y^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $4 x^{2} + y^{2} = 1$ with $- \sqrt{14 + 3 y^{2}}$ .

$4 {(- \sqrt{14 + 3 y^{2}})}^{2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $4 {(- \sqrt{14 + 3 y^{2}})}^{2} + y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{14 + 3 y^{2}}$ .

$4 ({(- 1)}^{2} {\sqrt{14 + 3 y^{2}}}^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$4 (1 {\sqrt{14 + 3 y^{2}}}^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{14 + 3 y^{2}}}^{2}$ by $1$ .

$4 {\sqrt{14 + 3 y^{2}}}^{2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{14 + 3 y^{2}}}^{2}$ as $14 + 3 y^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14 + 3 y^{2}}$ as ${(14 + 3 y^{2})}^{\frac{1}{2}}$ .

$4 {({(14 + 3 y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(14 + 3 y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4.3

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

$4 {(14 + 3 y^{2})}^{\frac{2}{2}} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$4 {(14 + 3 y^{2})}^{\frac{2}{2}} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$4 (14 + 3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.5

Apply the distributive property.

$4 \cdot 14 + 4 (3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.6

Multiply $4$ by $14$ .

$56 + 4 (3 y^{2}) + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.1.7

Multiply $3$ by $4$ .

$56 + 12 y^{2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$56 + 12 y^{2} + y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.1.2.1.2

Add $12 y^{2}$ and $y^{2}$ .

$56 + 13 y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

$56 + 13 y^{2} = 1$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2

Solve for $y$ in $56 + 13 y^{2} = 1$ .

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Step 3.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.1.1

Subtract $56$ from both sides of the equation.

$13 y^{2} = 1 - 56$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.1.2

Subtract $56$ from $1$ .

$13 y^{2} = - 55$

$x = - \sqrt{14 + 3 y^{2}}$

$13 y^{2} = - 55$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.2

Divide each term in $13 y^{2} = - 55$ by $13$ and simplify.

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Step 3.2.2.1

Divide each term in $13 y^{2} = - 55$ by $13$ .

$\frac{13 y^{2}}{13} = \frac{- 55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $13$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{13 y^{2}}{13} = \frac{- 55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y^{2} = \frac{- 55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y^{2} = \frac{- 55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y^{2} = - \frac{55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y^{2} = - \frac{55}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{55}{13}}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4

Simplify $\pm \sqrt{- \frac{55}{13}}$ .

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Step 3.2.4.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} (\frac{55}{13})}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.2

Pull terms out from under the radical.

$y = \pm i \sqrt{\frac{55}{13}}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.3

Rewrite $\sqrt{\frac{55}{13}}$ as $\frac{\sqrt{55}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55}}{\sqrt{13}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.4

Multiply $\frac{\sqrt{55}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5

Combine and simplify the denominator.

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Step 3.2.4.5.1

Multiply $\frac{\sqrt{55}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.2

Raise $\sqrt{13}$ to the power of $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.3

Raise $\sqrt{13}$ to the power of $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{\sqrt{13} \sqrt{13}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.4

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

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{\sqrt{13}}^{1 + 1}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.5

Add $1$ and $1$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{\sqrt{13}}^{2}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 3.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6.3

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

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{2}{2}}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.5.6.4.1

Cancel the common factor.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13^{\frac{2}{2}}})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6.4.2

Rewrite the expression.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.5.6.5

Evaluate the exponent.

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{55} \sqrt{13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.6

Simplify the numerator.

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Step 3.2.4.6.1

Combine using the product rule for radicals.

$y = \pm i (\frac{\sqrt{55 \cdot 13}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.6.2

Multiply $55$ by $13$ .

$y = \pm i (\frac{\sqrt{715}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \pm i (\frac{\sqrt{715}}{13})$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.4.7

Combine $i$ and $\frac{\sqrt{715}}{13}$ .

$y = \pm \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \pm \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

$y = \frac{i \sqrt{715}}{13}, - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 y^{2}}$

Step 3.3

Replace all occurrences of $y$ with $\frac{i \sqrt{715}}{13}$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{14 + 3 y^{2}}$ with $\frac{i \sqrt{715}}{13}$ .

$x = - \sqrt{14 + 3 {(\frac{i \sqrt{715}}{13})}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{14 + 3 {(\frac{i \sqrt{715}}{13})}^{2}}$ .

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Step 3.3.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.1.1

Apply the product rule to $\frac{i \sqrt{715}}{13}$ .

$x = - \sqrt{14 + 3 (\frac{{(i \sqrt{715})}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.1.2

Apply the product rule to $i \sqrt{715}$ .

$x = - \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2

Simplify the numerator.

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Step 3.3.2.1.2.1

Rewrite $i^{2}$ as $- 1$ .

$x = - \sqrt{14 + 3 (\frac{- 1 {\sqrt{715}}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2

Rewrite ${\sqrt{715}}^{2}$ as $715$ .

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Step 3.3.2.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{715}$ as $715^{\frac{1}{2}}$ .

$x = - \sqrt{14 + 3 (\frac{- 1 {(715^{\frac{1}{2}})}^{2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{1}{2} \cdot 2}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2.3

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

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.2.4.1

Cancel the common factor.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2.4.2

Rewrite the expression.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.2.2.5

Evaluate the exponent.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1.3.1

Raise $13$ to the power of $2$ .

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.2

Multiply $- 1$ by $715$ .

$x = - \sqrt{14 + 3 (\frac{- 715}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.3

Cancel the common factor of $- 715$ and $169$ .

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Step 3.3.2.1.3.3.1

Factor $13$ out of $- 715$ .

$x = - \sqrt{14 + 3 (\frac{13 (- 55)}{169})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.3.2

Cancel the common factors.

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Step 3.3.2.1.3.3.2.1

Factor $13$ out of $169$ .

$x = - \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.3.2.2

Cancel the common factor.

$x = - \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.3.2.3

Rewrite the expression.

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.3.4

Move the negative in front of the fraction.

$x = - \sqrt{14 + 3 (- \frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (- \frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.4

Multiply $3 (- \frac{55}{13})$ .

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Step 3.3.2.1.4.1

Multiply $- 1$ by $3$ .

$x = - \sqrt{14 - 3 (\frac{55}{13})}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.4.2

Combine $- 3$ and $\frac{55}{13}$ .

$x = - \sqrt{14 + \frac{- 3 \cdot 55}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.4.3

Multiply $- 3$ by $55$ .

$x = - \sqrt{14 + \frac{- 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + \frac{- 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.5

Move the negative in front of the fraction.

$x = - \sqrt{14 - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.6

To write $14$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = - \sqrt{14 \cdot \frac{13}{13} - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.7

Combine $14$ and $\frac{13}{13}$ .

$x = - \sqrt{\frac{14 \cdot 13}{13} - \frac{165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.8

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{14 \cdot 13 - 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.9

Simplify the numerator.

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Step 3.3.2.1.9.1

Multiply $14$ by $13$ .

$x = - \sqrt{\frac{182 - 165}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.9.2

Subtract $165$ from $182$ .

$x = - \sqrt{\frac{17}{13}}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \sqrt{\frac{17}{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.10

Rewrite $\sqrt{\frac{17}{13}}$ as $\frac{\sqrt{17}}{\sqrt{13}}$ .

$x = - \frac{\sqrt{17}}{\sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.11

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = - (\frac{\sqrt{17}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12

Combine and simplify the denominator.

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Step 3.3.2.1.12.1

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.2

Raise $\sqrt{13}$ to the power of $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.3

Raise $\sqrt{13}$ to the power of $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.4

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

$x = - \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.5

Add $1$ and $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 3.3.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6.3

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

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.12.6.4.1

Cancel the common factor.

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6.4.2

Rewrite the expression.

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.12.6.5

Evaluate the exponent.

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.13

Simplify the numerator.

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Step 3.3.2.1.13.1

Combine using the product rule for radicals.

$x = - \frac{\sqrt{17 \cdot 13}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.3.2.1.13.2

Multiply $17$ by $13$ .

$x = - \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = \frac{i \sqrt{715}}{13}$

Step 3.4

Replace all occurrences of $y$ with $- \frac{i \sqrt{715}}{13}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \sqrt{14 + 3 y^{2}}$ with $- \frac{i \sqrt{715}}{13}$ .

$x = - \sqrt{14 + 3 {(- \frac{i \sqrt{715}}{13})}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{14 + 3 {(- \frac{i \sqrt{715}}{13})}^{2}}$ .

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Step 3.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.2.1.1.1

Apply the product rule to $- \frac{i \sqrt{715}}{13}$ .

$x = - \sqrt{14 + 3 ({(- 1)}^{2} {(\frac{i \sqrt{715}}{13})}^{2})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.1.2

Apply the product rule to $\frac{i \sqrt{715}}{13}$ .

$x = - \sqrt{14 + 3 ({(- 1)}^{2} (\frac{{(i \sqrt{715})}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.1.3

Apply the product rule to $i \sqrt{715}$ .

$x = - \sqrt{14 + 3 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Raise $- 1$ to the power of $2$ .

$x = - \sqrt{14 + 3 (1 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}))}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.2.2

Multiply $\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}}$ by $1$ .

$x = - \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{i^{2} {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3

Simplify the numerator.

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Step 3.4.2.1.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = - \sqrt{14 + 3 (\frac{- 1 {\sqrt{715}}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2

Rewrite ${\sqrt{715}}^{2}$ as $715$ .

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Step 3.4.2.1.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{715}$ as $715^{\frac{1}{2}}$ .

$x = - \sqrt{14 + 3 (\frac{- 1 {(715^{\frac{1}{2}})}^{2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{1}{2} \cdot 2}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2.3

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

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.3.2.4.1

Cancel the common factor.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715^{\frac{2}{2}}}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2.4.2

Rewrite the expression.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.3.2.5

Evaluate the exponent.

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{13^{2}})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 3.4.2.1.4.1

Raise $13$ to the power of $2$ .

$x = - \sqrt{14 + 3 (\frac{- 1 \cdot 715}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.2

Multiply $- 1$ by $715$ .

$x = - \sqrt{14 + 3 (\frac{- 715}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.3

Cancel the common factor of $- 715$ and $169$ .

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Step 3.4.2.1.4.3.1

Factor $13$ out of $- 715$ .

$x = - \sqrt{14 + 3 (\frac{13 (- 55)}{169})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.3.2

Cancel the common factors.

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Step 3.4.2.1.4.3.2.1

Factor $13$ out of $169$ .

$x = - \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.3.2.2

Cancel the common factor.

$x = - \sqrt{14 + 3 (\frac{13 \cdot - 55}{13 \cdot 13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.3.2.3

Rewrite the expression.

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (\frac{- 55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.4.4

Move the negative in front of the fraction.

$x = - \sqrt{14 + 3 (- \frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + 3 (- \frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.5

Multiply $3 (- \frac{55}{13})$ .

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Step 3.4.2.1.5.1

Multiply $- 1$ by $3$ .

$x = - \sqrt{14 - 3 (\frac{55}{13})}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.5.2

Combine $- 3$ and $\frac{55}{13}$ .

$x = - \sqrt{14 + \frac{- 3 \cdot 55}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.5.3

Multiply $- 3$ by $55$ .

$x = - \sqrt{14 + \frac{- 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{14 + \frac{- 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.6

Move the negative in front of the fraction.

$x = - \sqrt{14 - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.7

To write $14$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = - \sqrt{14 \cdot \frac{13}{13} - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.8

Combine $14$ and $\frac{13}{13}$ .

$x = - \sqrt{\frac{14 \cdot 13}{13} - \frac{165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.9

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{14 \cdot 13 - 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.10

Simplify the numerator.

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Step 3.4.2.1.10.1

Multiply $14$ by $13$ .

$x = - \sqrt{\frac{182 - 165}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.10.2

Subtract $165$ from $182$ .

$x = - \sqrt{\frac{17}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \sqrt{\frac{17}{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.11

Rewrite $\sqrt{\frac{17}{13}}$ as $\frac{\sqrt{17}}{\sqrt{13}}$ .

$x = - \frac{\sqrt{17}}{\sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.12

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = - (\frac{\sqrt{17}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13

Combine and simplify the denominator.

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Step 3.4.2.1.13.1

Multiply $\frac{\sqrt{17}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.2

Raise $\sqrt{13}$ to the power of $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.3

Raise $\sqrt{13}$ to the power of $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.4

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

$x = - \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.5

Add $1$ and $1$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{{\sqrt{13}}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 3.4.2.1.13.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6.3

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

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.13.6.4.1

Cancel the common factor.

$x = - \frac{\sqrt{17} \sqrt{13}}{13^{\frac{2}{2}}}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6.4.2

Rewrite the expression.

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.13.6.5

Evaluate the exponent.

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{17} \sqrt{13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.14

Simplify the numerator.

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Step 3.4.2.1.14.1

Combine using the product rule for radicals.

$x = - \frac{\sqrt{17 \cdot 13}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 3.4.2.1.14.2

Multiply $17$ by $13$ .

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}$

$y = - \frac{i \sqrt{715}}{13}$

Step 4

List all of the solutions.

$x = \frac{\sqrt{221}}{13}, y = \frac{i \sqrt{715}}{13}$

$x = \frac{\sqrt{221}}{13}, y = - \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}, y = \frac{i \sqrt{715}}{13}$

$x = - \frac{\sqrt{221}}{13}, y = - \frac{i \sqrt{715}}{13}$

Step 5