Finite Math Examples

$x^{2} - y^{2} = 21$ , $2 x^{2} + y^{2} = 79$

Step 1

Solve for $x$ in $x^{2} - y^{2} = 21$ .

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Step 1.1

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

$x^{2} = 21 + y^{2}$

$2 x^{2} + y^{2} = 79$

Step 1.2

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

$x = \pm \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

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.

$x = \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

Step 1.3.2

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

$x = - \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

Step 1.3.3

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

$x = \sqrt{21 + y^{2}}$

$x = - \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$x = - \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$x = - \sqrt{21 + y^{2}}$

$2 x^{2} + y^{2} = 79$

Step 2

Solve the system $x = \sqrt{21 + y^{2}}, 2 x^{2} + y^{2} = 79$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{21 + y^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $2 x^{2} + y^{2} = 79$ with $\sqrt{21 + y^{2}}$ .

$2 {(\sqrt{21 + y^{2}})}^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $2 {(\sqrt{21 + y^{2}})}^{2} + y^{2}$ .

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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 ${\sqrt{21 + y^{2}}}^{2}$ as $21 + y^{2}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 + y^{2}}$ as ${(21 + y^{2})}^{\frac{1}{2}}$ .

$2 {({(21 + y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.1.2

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

$2 {(21 + y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.1.3

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

$2 {(21 + y^{2})}^{\frac{2}{2}} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

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.

$2 {(21 + y^{2})}^{\frac{2}{2}} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$2 (21 + y^{2}) + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$2 (21 + y^{2}) + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.1.5

Simplify.

$2 (21 + y^{2}) + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$2 (21 + y^{2}) + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.2

Apply the distributive property.

$2 \cdot 21 + 2 y^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.1.3

Multiply $2$ by $21$ .

$42 + 2 y^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$42 + 2 y^{2} + y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.1.2.1.2

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

$42 + 3 y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = \sqrt{21 + y^{2}}$

Step 2.2

Solve for $y$ in $42 + 3 y^{2} = 79$ .

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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 $42$ from both sides of the equation.

$3 y^{2} = 79 - 42$

$x = \sqrt{21 + y^{2}}$

Step 2.2.1.2

Subtract $42$ from $79$ .

$3 y^{2} = 37$

$x = \sqrt{21 + y^{2}}$

$3 y^{2} = 37$

$x = \sqrt{21 + y^{2}}$

Step 2.2.2

Divide each term in $3 y^{2} = 37$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y^{2} = 37$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.2.2.1.2

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

$y^{2} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.3

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

$y = \pm \sqrt{\frac{37}{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4

Simplify $\pm \sqrt{\frac{37}{3}}$ .

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Step 2.2.4.1

Rewrite $\sqrt{\frac{37}{3}}$ as $\frac{\sqrt{37}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37}}{\sqrt{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.2

Multiply $\frac{\sqrt{37}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3

Combine and simplify the denominator.

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Step 2.2.4.3.1

Multiply $\frac{\sqrt{37}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.4

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.4.3.6.1

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6.2

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6.3

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.4

Simplify the numerator.

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Step 2.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{37 \cdot 3}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.4.4.2

Multiply $37$ by $3$ .

$y = \pm \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

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.

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.5.2

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

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.2.5.3

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

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + y^{2}}$

Step 2.3

Replace all occurrences of $y$ with $\frac{\sqrt{111}}{3}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{21 + y^{2}}$ with $\frac{\sqrt{111}}{3}$ .

$x = \sqrt{21 + {(\frac{\sqrt{111}}{3})}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{21 + {(\frac{\sqrt{111}}{3})}^{2}}$ .

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Step 2.3.2.1.1

Apply the product rule to $\frac{\sqrt{111}}{3}$ .

$x = \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

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Step 2.3.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$x = \sqrt{21 + \frac{{(111^{\frac{1}{2}})}^{2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2.2

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

$x = \sqrt{21 + \frac{111^{\frac{1}{2} \cdot 2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2.3

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

$x = \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.4.1

Cancel the common factor.

$x = \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2.4.2

Rewrite the expression.

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.2.5

Evaluate the exponent.

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.3

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

$x = \sqrt{21 + \frac{111}{9}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.4

Cancel the common factor of $111$ and $9$ .

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Step 2.3.2.1.4.1

Factor $3$ out of $111$ .

$x = \sqrt{21 + \frac{3 (37)}{9}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.4.2

Cancel the common factors.

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Step 2.3.2.1.4.2.1

Factor $3$ out of $9$ .

$x = \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.4.2.2

Cancel the common factor.

$x = \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.4.2.3

Rewrite the expression.

$x = \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.5

To write $21$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \sqrt{21 \cdot \frac{3}{3} + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.6

Combine $21$ and $\frac{3}{3}$ .

$x = \sqrt{\frac{21 \cdot 3}{3} + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.7

Combine the numerators over the common denominator.

$x = \sqrt{\frac{21 \cdot 3 + 37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.8

Simplify the numerator.

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Step 2.3.2.1.8.1

Multiply $21$ by $3$ .

$x = \sqrt{\frac{63 + 37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.8.2

Add $63$ and $37$ .

$x = \sqrt{\frac{100}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = \sqrt{\frac{100}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.9

Rewrite $\sqrt{\frac{100}{3}}$ as $\frac{\sqrt{100}}{\sqrt{3}}$ .

$x = \frac{\sqrt{100}}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.10

Simplify the numerator.

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Step 2.3.2.1.10.1

Rewrite $100$ as $10^{2}$ .

$x = \frac{\sqrt{10^{2}}}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.10.2

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

$x = \frac{10}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.11

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12

Combine and simplify the denominator.

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Step 2.3.2.1.12.1

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.4

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

$x = \frac{10 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.5

Add $1$ and $1$ .

$x = \frac{10 \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.2.1.12.6.1

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

$x = \frac{10 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.6.2

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

$x = \frac{10 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.6.3

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

$x = \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \frac{\sqrt{111}}{3}$

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.

$x = \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.6.4.2

Rewrite the expression.

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

Step 2.3.2.1.12.6.5

Evaluate the exponent.

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

Step 2.4

Replace all occurrences of $y$ with $- \frac{\sqrt{111}}{3}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{21 + y^{2}}$ with $- \frac{\sqrt{111}}{3}$ .

$x = \sqrt{21 + {(- \frac{\sqrt{111}}{3})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{21 + {(- \frac{\sqrt{111}}{3})}^{2}}$ .

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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 $- \frac{\sqrt{111}}{3}$ .

$x = \sqrt{21 + {(- 1)}^{2} {(\frac{\sqrt{111}}{3})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.1.2

Apply the product rule to $\frac{\sqrt{111}}{3}$ .

$x = \sqrt{21 + {(- 1)}^{2} (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + {(- 1)}^{2} (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

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 $2$ .

$x = \sqrt{21 + 1 (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.2.2

Multiply $\frac{{\sqrt{111}}^{2}}{3^{2}}$ by $1$ .

$x = \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

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Step 2.4.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$x = \sqrt{21 + \frac{{(111^{\frac{1}{2}})}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3.2

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

$x = \sqrt{21 + \frac{111^{\frac{1}{2} \cdot 2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3.3

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

$x = \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.3.4.1

Cancel the common factor.

$x = \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3.4.2

Rewrite the expression.

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.3.5

Evaluate the exponent.

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.4

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

$x = \sqrt{21 + \frac{111}{9}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.5

Cancel the common factor of $111$ and $9$ .

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Step 2.4.2.1.5.1

Factor $3$ out of $111$ .

$x = \sqrt{21 + \frac{3 (37)}{9}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.5.2

Cancel the common factors.

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Step 2.4.2.1.5.2.1

Factor $3$ out of $9$ .

$x = \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.5.2.2

Cancel the common factor.

$x = \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.5.2.3

Rewrite the expression.

$x = \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.6

To write $21$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \sqrt{21 \cdot \frac{3}{3} + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.7

Combine $21$ and $\frac{3}{3}$ .

$x = \sqrt{\frac{21 \cdot 3}{3} + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.8

Combine the numerators over the common denominator.

$x = \sqrt{\frac{21 \cdot 3 + 37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.9

Simplify the numerator.

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Step 2.4.2.1.9.1

Multiply $21$ by $3$ .

$x = \sqrt{\frac{63 + 37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.9.2

Add $63$ and $37$ .

$x = \sqrt{\frac{100}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \sqrt{\frac{100}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.10

Rewrite $\sqrt{\frac{100}{3}}$ as $\frac{\sqrt{100}}{\sqrt{3}}$ .

$x = \frac{\sqrt{100}}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.11

Simplify the numerator.

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Step 2.4.2.1.11.1

Rewrite $100$ as $10^{2}$ .

$x = \frac{\sqrt{10^{2}}}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.11.2

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

$x = \frac{10}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.12

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13

Combine and simplify the denominator.

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Step 2.4.2.1.13.1

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.4

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

$x = \frac{10 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.5

Add $1$ and $1$ .

$x = \frac{10 \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.2.1.13.6.1

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

$x = \frac{10 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.6.2

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

$x = \frac{10 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.6.3

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

$x = \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \frac{\sqrt{111}}{3}$

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.

$x = \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.6.4.2

Rewrite the expression.

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

Step 2.4.2.1.13.6.5

Evaluate the exponent.

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

Step 3

Solve the system $x = - \sqrt{21 + y^{2}}, 2 x^{2} + y^{2} = 79$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{21 + y^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $2 x^{2} + y^{2} = 79$ with $- \sqrt{21 + y^{2}}$ .

$2 {(- \sqrt{21 + y^{2}})}^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $2 {(- \sqrt{21 + y^{2}})}^{2} + y^{2}$ .

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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 $- \sqrt{21 + y^{2}}$ .

$2 ({(- 1)}^{2} {\sqrt{21 + y^{2}}}^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.2

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

$2 (1 {\sqrt{21 + y^{2}}}^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{21 + y^{2}}}^{2}$ by $1$ .

$2 {\sqrt{21 + y^{2}}}^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{21 + y^{2}}}^{2}$ as $21 + y^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 + y^{2}}$ as ${(21 + y^{2})}^{\frac{1}{2}}$ .

$2 {({(21 + y^{2})}^{\frac{1}{2}})}^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.4.2

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

$2 {(21 + y^{2})}^{\frac{1}{2} \cdot 2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.4.3

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

$2 {(21 + y^{2})}^{\frac{2}{2}} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

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.

$2 {(21 + y^{2})}^{\frac{2}{2}} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$2 (21 + y^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$2 (21 + y^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$2 (21 + y^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$2 (21 + y^{2}) + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.5

Apply the distributive property.

$2 \cdot 21 + 2 y^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.1.6

Multiply $2$ by $21$ .

$42 + 2 y^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$42 + 2 y^{2} + y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.1.2.1.2

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

$42 + 3 y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

$42 + 3 y^{2} = 79$

$x = - \sqrt{21 + y^{2}}$

Step 3.2

Solve for $y$ in $42 + 3 y^{2} = 79$ .

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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 $42$ from both sides of the equation.

$3 y^{2} = 79 - 42$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.1.2

Subtract $42$ from $79$ .

$3 y^{2} = 37$

$x = - \sqrt{21 + y^{2}}$

$3 y^{2} = 37$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.2

Divide each term in $3 y^{2} = 37$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 y^{2} = 37$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.2.2.1.2

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

$y^{2} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

$y^{2} = \frac{37}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.3

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

$y = \pm \sqrt{\frac{37}{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4

Simplify $\pm \sqrt{\frac{37}{3}}$ .

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Step 3.2.4.1

Rewrite $\sqrt{\frac{37}{3}}$ as $\frac{\sqrt{37}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37}}{\sqrt{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.2

Multiply $\frac{\sqrt{37}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3

Combine and simplify the denominator.

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Step 3.2.4.3.1

Multiply $\frac{\sqrt{37}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.4

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.4.3.6.1

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6.2

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6.3

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

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{37} \sqrt{3}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.4

Simplify the numerator.

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Step 3.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{37 \cdot 3}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.4.4.2

Multiply $37$ by $3$ .

$y = \pm \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \pm \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

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.

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.5.2

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

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.2.5.3

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

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

$y = \frac{\sqrt{111}}{3}, - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + y^{2}}$

Step 3.3

Replace all occurrences of $y$ with $\frac{\sqrt{111}}{3}$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{21 + y^{2}}$ with $\frac{\sqrt{111}}{3}$ .

$x = - \sqrt{21 + {(\frac{\sqrt{111}}{3})}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{21 + {(\frac{\sqrt{111}}{3})}^{2}}$ .

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Step 3.3.2.1.1

Apply the product rule to $\frac{\sqrt{111}}{3}$ .

$x = - \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

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Step 3.3.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$x = - \sqrt{21 + \frac{{(111^{\frac{1}{2}})}^{2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2.2

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

$x = - \sqrt{21 + \frac{111^{\frac{1}{2} \cdot 2}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2.3

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

$x = - \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.4.1

Cancel the common factor.

$x = - \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2.4.2

Rewrite the expression.

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.2.5

Evaluate the exponent.

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.3

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

$x = - \sqrt{21 + \frac{111}{9}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.4

Cancel the common factor of $111$ and $9$ .

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Step 3.3.2.1.4.1

Factor $3$ out of $111$ .

$x = - \sqrt{21 + \frac{3 (37)}{9}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.4.2

Cancel the common factors.

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Step 3.3.2.1.4.2.1

Factor $3$ out of $9$ .

$x = - \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.4.2.2

Cancel the common factor.

$x = - \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.4.2.3

Rewrite the expression.

$x = - \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.5

To write $21$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \sqrt{21 \cdot \frac{3}{3} + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.6

Combine $21$ and $\frac{3}{3}$ .

$x = - \sqrt{\frac{21 \cdot 3}{3} + \frac{37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.7

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{21 \cdot 3 + 37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.8

Simplify the numerator.

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Step 3.3.2.1.8.1

Multiply $21$ by $3$ .

$x = - \sqrt{\frac{63 + 37}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.8.2

Add $63$ and $37$ .

$x = - \sqrt{\frac{100}{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \sqrt{\frac{100}{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.9

Rewrite $\sqrt{\frac{100}{3}}$ as $\frac{\sqrt{100}}{\sqrt{3}}$ .

$x = - \frac{\sqrt{100}}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.10

Simplify the numerator.

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Step 3.3.2.1.10.1

Rewrite $100$ as $10^{2}$ .

$x = - \frac{\sqrt{10^{2}}}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.10.2

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

$x = - \frac{10}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10}{\sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.11

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - (\frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12

Combine and simplify the denominator.

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Step 3.3.2.1.12.1

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.4

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

$x = - \frac{10 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.5

Add $1$ and $1$ .

$x = - \frac{10 \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.2.1.12.6.1

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

$x = - \frac{10 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.6.2

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

$x = - \frac{10 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.6.3

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

$x = - \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \frac{\sqrt{111}}{3}$

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.

$x = - \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.6.4.2

Rewrite the expression.

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

Step 3.3.2.1.12.6.5

Evaluate the exponent.

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = \frac{\sqrt{111}}{3}$

Step 3.4

Replace all occurrences of $y$ with $- \frac{\sqrt{111}}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \sqrt{21 + y^{2}}$ with $- \frac{\sqrt{111}}{3}$ .

$x = - \sqrt{21 + {(- \frac{\sqrt{111}}{3})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{21 + {(- \frac{\sqrt{111}}{3})}^{2}}$ .

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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 $- \frac{\sqrt{111}}{3}$ .

$x = - \sqrt{21 + {(- 1)}^{2} {(\frac{\sqrt{111}}{3})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.1.2

Apply the product rule to $\frac{\sqrt{111}}{3}$ .

$x = - \sqrt{21 + {(- 1)}^{2} (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + {(- 1)}^{2} (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

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 $2$ .

$x = - \sqrt{21 + 1 (\frac{{\sqrt{111}}^{2}}{3^{2}})}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.2.2

Multiply $\frac{{\sqrt{111}}^{2}}{3^{2}}$ by $1$ .

$x = - \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{{\sqrt{111}}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

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Step 3.4.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$x = - \sqrt{21 + \frac{{(111^{\frac{1}{2}})}^{2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3.2

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

$x = - \sqrt{21 + \frac{111^{\frac{1}{2} \cdot 2}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3.3

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

$x = - \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.3.4.1

Cancel the common factor.

$x = - \sqrt{21 + \frac{111^{\frac{2}{2}}}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3.4.2

Rewrite the expression.

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.3.5

Evaluate the exponent.

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{111}{3^{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.4

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

$x = - \sqrt{21 + \frac{111}{9}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.5

Cancel the common factor of $111$ and $9$ .

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Step 3.4.2.1.5.1

Factor $3$ out of $111$ .

$x = - \sqrt{21 + \frac{3 (37)}{9}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.5.2

Cancel the common factors.

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Step 3.4.2.1.5.2.1

Factor $3$ out of $9$ .

$x = - \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.5.2.2

Cancel the common factor.

$x = - \sqrt{21 + \frac{3 \cdot 37}{3 \cdot 3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.5.2.3

Rewrite the expression.

$x = - \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{21 + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.6

To write $21$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \sqrt{21 \cdot \frac{3}{3} + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.7

Combine $21$ and $\frac{3}{3}$ .

$x = - \sqrt{\frac{21 \cdot 3}{3} + \frac{37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.8

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{21 \cdot 3 + 37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.9

Simplify the numerator.

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Step 3.4.2.1.9.1

Multiply $21$ by $3$ .

$x = - \sqrt{\frac{63 + 37}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.9.2

Add $63$ and $37$ .

$x = - \sqrt{\frac{100}{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \sqrt{\frac{100}{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.10

Rewrite $\sqrt{\frac{100}{3}}$ as $\frac{\sqrt{100}}{\sqrt{3}}$ .

$x = - \frac{\sqrt{100}}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.11

Simplify the numerator.

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Step 3.4.2.1.11.1

Rewrite $100$ as $10^{2}$ .

$x = - \frac{\sqrt{10^{2}}}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.11.2

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

$x = - \frac{10}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10}{\sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.12

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - (\frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13

Combine and simplify the denominator.

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Step 3.4.2.1.13.1

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = - \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.4

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

$x = - \frac{10 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.5

Add $1$ and $1$ .

$x = - \frac{10 \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.2.1.13.6.1

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

$x = - \frac{10 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.6.2

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

$x = - \frac{10 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.6.3

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

$x = - \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \frac{\sqrt{111}}{3}$

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.

$x = - \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.6.4.2

Rewrite the expression.

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

Step 3.4.2.1.13.6.5

Evaluate the exponent.

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}$

$y = - \frac{\sqrt{111}}{3}$

Step 4

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

$(\frac{10 \sqrt{3}}{3}, \frac{\sqrt{111}}{3})$

$(\frac{10 \sqrt{3}}{3}, - \frac{\sqrt{111}}{3})$

$(- \frac{10 \sqrt{3}}{3}, \frac{\sqrt{111}}{3})$

$(- \frac{10 \sqrt{3}}{3}, - \frac{\sqrt{111}}{3})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{10 \sqrt{3}}{3}, \frac{\sqrt{111}}{3}), (\frac{10 \sqrt{3}}{3}, - \frac{\sqrt{111}}{3}), (- \frac{10 \sqrt{3}}{3}, \frac{\sqrt{111}}{3}), (- \frac{10 \sqrt{3}}{3}, - \frac{\sqrt{111}}{3})$

Equation Form:

$x = \frac{10 \sqrt{3}}{3}, y = \frac{\sqrt{111}}{3}$

$x = \frac{10 \sqrt{3}}{3}, y = - \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}, y = \frac{\sqrt{111}}{3}$

$x = - \frac{10 \sqrt{3}}{3}, y = - \frac{\sqrt{111}}{3}$

Step 6