Algebra Examples

$4 x^{2} - 3 y^{2} = - 11$ , $5 x^{2} + 2 y^{2} = 38$

Step 1

Solve for $x$ in $4 x^{2} - 3 y^{2} = - 11$ .

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Step 1.1

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

$4 x^{2} = - 11 + 3 y^{2}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.2

Divide each term in $4 x^{2} = - 11 + 3 y^{2}$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x^{2} = - 11 + 3 y^{2}$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{- 11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{- 11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.2.2.1.2

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

$x^{2} = \frac{- 11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

$x^{2} = \frac{- 11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

$x^{2} = \frac{- 11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

$x^{2} = - \frac{11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

$x^{2} = - \frac{11}{4} + \frac{3 y^{2}}{4}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.3

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

$x = \pm \sqrt{- \frac{11}{4} + \frac{3 y^{2}}{4}}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4

Simplify $\pm \sqrt{- \frac{11}{4} + \frac{3 y^{2}}{4}}$ .

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Step 1.4.1

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- 11 + 3 y^{2}}{4}}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4.2

Rewrite $\frac{- 11 + 3 y^{2}}{4}$ as ${(\frac{1}{2})}^{2} (- 11 + 3 y^{2})$ .

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Step 1.4.2.1

Factor the perfect power $1^{2}$ out of $- 11 + 3 y^{2}$ .

$x = \pm \sqrt{\frac{1^{2} (- 11 + 3 y^{2})}{4}}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4.2.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} (- 11 + 3 y^{2})}{2^{2} \cdot 1}}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4.2.3

Rearrange the fraction $\frac{1^{2} (- 11 + 3 y^{2})}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} (- 11 + 3 y^{2})}$

$5 x^{2} + 2 y^{2} = 38$

$x = \pm \sqrt{{(\frac{1}{2})}^{2} (- 11 + 3 y^{2})}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4.3

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \cdot \sqrt{- 11 + 3 y^{2}}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.4.4

Combine $\frac{1}{2}$ and $\sqrt{- 11 + 3 y^{2}}$ .

$x = \pm \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

$x = \pm \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.5

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

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Step 1.5.1

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

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.5.2

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

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

Step 1.5.3

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

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 x^{2} + 2 y^{2} = 38$

Step 2

Solve the system $x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}, 5 x^{2} + 2 y^{2} = 38$ .

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Step 2.1

Replace all occurrences of $x$ with $\frac{\sqrt{- 11 + 3 y^{2}}}{2}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $5 x^{2} + 2 y^{2} = 38$ with $\frac{\sqrt{- 11 + 3 y^{2}}}{2}$ .

$5 {(\frac{\sqrt{- 11 + 3 y^{2}}}{2})}^{2} + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $5 {(\frac{\sqrt{- 11 + 3 y^{2}}}{2})}^{2} + 2 y^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Apply the product rule to $\frac{\sqrt{- 11 + 3 y^{2}}}{2}$ .

$5 (\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2

Rewrite ${\sqrt{- 11 + 3 y^{2}}}^{2}$ as $- 11 + 3 y^{2}$ .

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Step 2.1.2.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 11 + 3 y^{2}}$ as ${(- 11 + 3 y^{2})}^{\frac{1}{2}}$ .

$5 (\frac{{({(- 11 + 3 y^{2})}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2.2

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

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2.3

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

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.2.4.1

Cancel the common factor.

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2.4.2

Rewrite the expression.

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.2.5

Simplify.

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.3

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

$5 (\frac{- 11 + 3 y^{2}}{4}) + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.1.4

Combine $5$ and $\frac{- 11 + 3 y^{2}}{4}$ .

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.2

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

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} \cdot \frac{4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.3

Simplify terms.

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Step 2.1.2.1.3.1

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

$\frac{5 (- 11 + 3 y^{2})}{4} + \frac{2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{5 (- 11 + 3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{5 (- 11 + 3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

Apply the distributive property.

$\frac{5 \cdot - 11 + 5 (3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.4.2

Multiply $5$ by $- 11$ .

$\frac{- 55 + 5 (3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.4.3

Multiply $3$ by $5$ .

$\frac{- 55 + 15 y^{2} + 2 y^{2} \cdot 4}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.4.4

Multiply $4$ by $2$ .

$\frac{- 55 + 15 y^{2} + 8 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.4.5

Add $15 y^{2}$ and $8 y^{2}$ .

$\frac{- 55 + 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{- 55 + 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.5

Simplify with factoring out.

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Step 2.1.2.1.5.1

Rewrite $- 55$ as $- 1 (55)$ .

$\frac{- 1 \cdot 55 + 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.5.2

Factor $- 1$ out of $23 y^{2}$ .

$\frac{- 1 \cdot 55 - (- 23 y^{2})}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.5.3

Factor $- 1$ out of $- 1 (55) - (- 23 y^{2})$ .

$\frac{- 1 (55 - 23 y^{2})}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.1.2.1.5.4

Move the negative in front of the fraction.

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2

Solve for $y$ in $- \frac{55 - 23 y^{2}}{4} = 38$ .

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Step 2.2.1

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{55 - 23 y^{2}}{4}) = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2

Simplify both sides of the equation.

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Step 2.2.2.1

Simplify the left side.

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Step 2.2.2.1.1

Simplify $- 4 (- \frac{55 - 23 y^{2}}{4})$ .

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Step 2.2.2.1.1.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1.1.1

Move the leading negative in $- \frac{55 - 23 y^{2}}{4}$ into the numerator.

$- 4 \frac{- (55 - 23 y^{2})}{4} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- (55 - 23 y^{2})}{4}) = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.1.1.1.3

Cancel the common factor.

$4 \cdot (- 1 \frac{- (55 - 23 y^{2})}{4}) = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.1.1.1.4

Rewrite the expression.

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.1.1.2

Multiply.

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Step 2.2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (55 - 23 y^{2}) = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.1.1.2.2

Multiply $55 - 23 y^{2}$ by $1$ .

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

Multiply $- 4$ by $38$ .

$55 - 23 y^{2} = - 152$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 152$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 152$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.3

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

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Step 2.2.3.1

Subtract $55$ from both sides of the equation.

$- 23 y^{2} = - 152 - 55$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.3.2

Subtract $55$ from $- 152$ .

$- 23 y^{2} = - 207$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- 23 y^{2} = - 207$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.4

Divide each term in $- 23 y^{2} = - 207$ by $- 23$ and simplify.

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Step 2.2.4.1

Divide each term in $- 23 y^{2} = - 207$ by $- 23$ .

$\frac{- 23 y^{2}}{- 23} = \frac{- 207}{- 23}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $- 23$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{- 23 y^{2}}{- 23} = \frac{- 207}{- 23}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.4.2.1.2

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

$y^{2} = \frac{- 207}{- 23}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = \frac{- 207}{- 23}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = \frac{- 207}{- 23}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Divide $- 207$ by $- 23$ .

$y^{2} = 9$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = 9$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = 9$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.5

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

$y = \pm \sqrt{9}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.6

Simplify $\pm \sqrt{9}$ .

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Step 2.2.6.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{3^{2}}$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.6.2

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

$y = \pm 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = \pm 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.7

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

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Step 2.2.7.1

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

$y = 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.7.2

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

$y = - 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.2.7.3

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

$y = 3, - 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = 3, - 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = 3, - 3$

$x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ with $3$ .

$x = \frac{\sqrt{- 11 + 3 {(3)}^{2}}}{2}$

$y = 3$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\frac{\sqrt{- 11 + 3 {(3)}^{2}}}{2}$ .

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Step 2.3.2.1.1

Simplify the numerator.

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Step 2.3.2.1.1.1

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 2.3.2.1.1.1.1

Multiply $3$ by $3^{2}$ .

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Step 2.3.2.1.1.1.1.1

Raise $3$ to the power of $1$ .

$x = \frac{\sqrt{- 11 + 3 \cdot 3^{2}}}{2}$

$y = 3$

Step 2.3.2.1.1.1.1.2

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

$x = \frac{\sqrt{- 11 + 3^{1 + 2}}}{2}$

$y = 3$

$x = \frac{\sqrt{- 11 + 3^{1 + 2}}}{2}$

$y = 3$

Step 2.3.2.1.1.1.2

Add $1$ and $2$ .

$x = \frac{\sqrt{- 11 + 3^{3}}}{2}$

$y = 3$

$x = \frac{\sqrt{- 11 + 3^{3}}}{2}$

$y = 3$

Step 2.3.2.1.1.2

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

$x = \frac{\sqrt{- 11 + 27}}{2}$

$y = 3$

Step 2.3.2.1.1.3

Add $- 11$ and $27$ .

$x = \frac{\sqrt{16}}{2}$

$y = 3$

Step 2.3.2.1.1.4

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2}}}{2}$

$y = 3$

Step 2.3.2.1.1.5

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

$x = \frac{4}{2}$

$y = 3$

$x = \frac{4}{2}$

$y = 3$

Step 2.3.2.1.2

Divide $4$ by $2$ .

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

Step 2.4

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

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Step 2.4.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ with $- 3$ .

$x = \frac{\sqrt{- 11 + 3 {(- 3)}^{2}}}{2}$

$y = - 3$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{\sqrt{- 11 + 3 {(- 3)}^{2}}}{2}$ .

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Step 2.4.2.1.1

Simplify the numerator.

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Step 2.4.2.1.1.1

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

$x = \frac{\sqrt{- 11 + 3 \cdot 9}}{2}$

$y = - 3$

Step 2.4.2.1.1.2

Multiply $3$ by $9$ .

$x = \frac{\sqrt{- 11 + 27}}{2}$

$y = - 3$

Step 2.4.2.1.1.3

Add $- 11$ and $27$ .

$x = \frac{\sqrt{16}}{2}$

$y = - 3$

Step 2.4.2.1.1.4

Rewrite $16$ as $4^{2}$ .

$x = \frac{\sqrt{4^{2}}}{2}$

$y = - 3$

Step 2.4.2.1.1.5

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

$x = \frac{4}{2}$

$y = - 3$

$x = \frac{4}{2}$

$y = - 3$

Step 2.4.2.1.2

Divide $4$ by $2$ .

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

Step 3

Solve the system $x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}, 5 x^{2} + 2 y^{2} = 38$ .

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Step 3.1

Replace all occurrences of $x$ with $- \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $5 x^{2} + 2 y^{2} = 38$ with $- \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ .

$5 {(- \frac{\sqrt{- 11 + 3 y^{2}}}{2})}^{2} + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $5 {(- \frac{\sqrt{- 11 + 3 y^{2}}}{2})}^{2} + 2 y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

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

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Step 3.1.2.1.1.1.1

Apply the product rule to $- \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ .

$5 ({(- 1)}^{2} {(\frac{\sqrt{- 11 + 3 y^{2}}}{2})}^{2}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.1.2

Apply the product rule to $\frac{\sqrt{- 11 + 3 y^{2}}}{2}$ .

$5 ({(- 1)}^{2} (\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}})) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 ({(- 1)}^{2} (\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}})) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.2

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

$5 (1 (\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}})) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.3

Multiply $\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}}$ by $1$ .

$5 (\frac{{\sqrt{- 11 + 3 y^{2}}}^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{- 11 + 3 y^{2}}}^{2}$ as $- 11 + 3 y^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 11 + 3 y^{2}}$ as ${(- 11 + 3 y^{2})}^{\frac{1}{2}}$ .

$5 (\frac{{({(- 11 + 3 y^{2})}^{\frac{1}{2}})}^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.4.2

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

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{1}{2} \cdot 2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.4.3

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

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

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.

$5 (\frac{{(- 11 + 3 y^{2})}^{\frac{2}{2}}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.4.5

Simplify.

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$5 (\frac{- 11 + 3 y^{2}}{2^{2}}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.5

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

$5 (\frac{- 11 + 3 y^{2}}{4}) + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.1.6

Combine $5$ and $\frac{- 11 + 3 y^{2}}{4}$ .

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.2

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

$\frac{5 (- 11 + 3 y^{2})}{4} + 2 y^{2} \cdot \frac{4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.3

Simplify terms.

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Step 3.1.2.1.3.1

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

$\frac{5 (- 11 + 3 y^{2})}{4} + \frac{2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{5 (- 11 + 3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{5 (- 11 + 3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.4

Simplify the numerator.

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Step 3.1.2.1.4.1

Apply the distributive property.

$\frac{5 \cdot - 11 + 5 (3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.4.2

Multiply $5$ by $- 11$ .

$\frac{- 55 + 5 (3 y^{2}) + 2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.4.3

Multiply $3$ by $5$ .

$\frac{- 55 + 15 y^{2} + 2 y^{2} \cdot 4}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.4.4

Multiply $4$ by $2$ .

$\frac{- 55 + 15 y^{2} + 8 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.4.5

Add $15 y^{2}$ and $8 y^{2}$ .

$\frac{- 55 + 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$\frac{- 55 + 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.5

Simplify with factoring out.

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Step 3.1.2.1.5.1

Rewrite $- 55$ as $- 1 (55)$ .

$\frac{- 1 \cdot 55 + 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.5.2

Factor $- 1$ out of $23 y^{2}$ .

$\frac{- 1 \cdot 55 - (- 23 y^{2})}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.5.3

Factor $- 1$ out of $- 1 (55) - (- 23 y^{2})$ .

$\frac{- 1 (55 - 23 y^{2})}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.1.2.1.5.4

Move the negative in front of the fraction.

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- \frac{55 - 23 y^{2}}{4} = 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2

Solve for $y$ in $- \frac{55 - 23 y^{2}}{4} = 38$ .

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Step 3.2.1

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{55 - 23 y^{2}}{4}) = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2

Simplify both sides of the equation.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Simplify $- 4 (- \frac{55 - 23 y^{2}}{4})$ .

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Step 3.2.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1.1.1

Move the leading negative in $- \frac{55 - 23 y^{2}}{4}$ into the numerator.

$- 4 \frac{- (55 - 23 y^{2})}{4} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- (55 - 23 y^{2})}{4}) = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.1.1.1.3

Cancel the common factor.

$4 \cdot (- 1 \frac{- (55 - 23 y^{2})}{4}) = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.1.1.1.4

Rewrite the expression.

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.1.1.2

Multiply.

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Step 3.2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (55 - 23 y^{2}) = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.1.1.2.2

Multiply $55 - 23 y^{2}$ by $1$ .

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 4 \cdot 38$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Multiply $- 4$ by $38$ .

$55 - 23 y^{2} = - 152$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 152$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$55 - 23 y^{2} = - 152$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.3

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

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Step 3.2.3.1

Subtract $55$ from both sides of the equation.

$- 23 y^{2} = - 152 - 55$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.3.2

Subtract $55$ from $- 152$ .

$- 23 y^{2} = - 207$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$- 23 y^{2} = - 207$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.4

Divide each term in $- 23 y^{2} = - 207$ by $- 23$ and simplify.

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Step 3.2.4.1

Divide each term in $- 23 y^{2} = - 207$ by $- 23$ .

$\frac{- 23 y^{2}}{- 23} = \frac{- 207}{- 23}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $- 23$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{- 23 y^{2}}{- 23} = \frac{- 207}{- 23}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.4.2.1.2

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

$y^{2} = \frac{- 207}{- 23}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = \frac{- 207}{- 23}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = \frac{- 207}{- 23}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.4.3

Simplify the right side.

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Step 3.2.4.3.1

Divide $- 207$ by $- 23$ .

$y^{2} = 9$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = 9$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y^{2} = 9$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.5

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

$y = \pm \sqrt{9}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.6

Simplify $\pm \sqrt{9}$ .

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Step 3.2.6.1

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{3^{2}}$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.6.2

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

$y = \pm 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = \pm 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.7

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

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Step 3.2.7.1

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

$y = 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.7.2

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

$y = - 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.2.7.3

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

$y = 3, - 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = 3, - 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

$y = 3, - 3$

$x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ with $3$ .

$x = - \frac{\sqrt{- 11 + 3 {(3)}^{2}}}{2}$

$y = 3$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{\sqrt{- 11 + 3 {(3)}^{2}}}{2}$ .

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Step 3.3.2.1.1

Simplify the numerator.

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Step 3.3.2.1.1.1

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 3.3.2.1.1.1.1

Multiply $3$ by $3^{2}$ .

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Step 3.3.2.1.1.1.1.1

Raise $3$ to the power of $1$ .

$x = - \frac{\sqrt{- 11 + 3 \cdot 3^{2}}}{2}$

$y = 3$

Step 3.3.2.1.1.1.1.2

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

$x = - \frac{\sqrt{- 11 + 3^{1 + 2}}}{2}$

$y = 3$

$x = - \frac{\sqrt{- 11 + 3^{1 + 2}}}{2}$

$y = 3$

Step 3.3.2.1.1.1.2

Add $1$ and $2$ .

$x = - \frac{\sqrt{- 11 + 3^{3}}}{2}$

$y = 3$

$x = - \frac{\sqrt{- 11 + 3^{3}}}{2}$

$y = 3$

Step 3.3.2.1.1.2

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

$x = - \frac{\sqrt{- 11 + 27}}{2}$

$y = 3$

Step 3.3.2.1.1.3

Add $- 11$ and $27$ .

$x = - \frac{\sqrt{16}}{2}$

$y = 3$

Step 3.3.2.1.1.4

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2}}}{2}$

$y = 3$

Step 3.3.2.1.1.5

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

$x = - \frac{4}{2}$

$y = 3$

$x = - \frac{4}{2}$

$y = 3$

Step 3.3.2.1.2

Simplify the expression.

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Step 3.3.2.1.2.1

Divide $4$ by $2$ .

$x = - 1 \cdot 2$

$y = 3$

Step 3.3.2.1.2.2

Multiply $- 1$ by $2$ .

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{- 11 + 3 y^{2}}}{2}$ with $- 3$ .

$x = - \frac{\sqrt{- 11 + 3 {(- 3)}^{2}}}{2}$

$y = - 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \frac{\sqrt{- 11 + 3 {(- 3)}^{2}}}{2}$ .

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

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

$x = - \frac{\sqrt{- 11 + 3 \cdot 9}}{2}$

$y = - 3$

Step 3.4.2.1.1.2

Multiply $3$ by $9$ .

$x = - \frac{\sqrt{- 11 + 27}}{2}$

$y = - 3$

Step 3.4.2.1.1.3

Add $- 11$ and $27$ .

$x = - \frac{\sqrt{16}}{2}$

$y = - 3$

Step 3.4.2.1.1.4

Rewrite $16$ as $4^{2}$ .

$x = - \frac{\sqrt{4^{2}}}{2}$

$y = - 3$

Step 3.4.2.1.1.5

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

$x = - \frac{4}{2}$

$y = - 3$

$x = - \frac{4}{2}$

$y = - 3$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Divide $4$ by $2$ .

$x = - 1 \cdot 2$

$y = - 3$

Step 3.4.2.1.2.2

Multiply $- 1$ by $2$ .

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

$x = - 2$

$y = - 3$

Step 4

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

$(2, 3)$

$(2, - 3)$

$(- 2, 3)$

$(- 2, - 3)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, 3), (2, - 3), (- 2, 3), (- 2, - 3)$

Equation Form:

$x = 2, y = 3$

$x = 2, y = - 3$

$x = - 2, y = 3$

$x = - 2, y = - 3$

Step 6