Finite Math Examples

$x^{2} + 2 y^{2} = 6$ , $2 x^{2} - 3 y^{2} = 5$

Step 1

Solve for $x$ in $x^{2} + 2 y^{2} = 6$ .

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Step 1.1

Subtract $2 y^{2}$ from both sides of the equation.

$x^{2} = 6 - 2 y^{2}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.2

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

$x = \pm \sqrt{6 - 2 y^{2}}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.3

Factor $2$ out of $6 - 2 y^{2}$ .

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Step 1.3.1

Factor $2$ out of $6$ .

$x = \pm \sqrt{2 (3) - 2 y^{2}}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.3.2

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

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

$2 x^{2} - 3 y^{2} = 5$

Step 1.3.3

Factor $2$ out of $2 (3) + 2 (- y^{2})$ .

$x = \pm \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

$x = \pm \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.4

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

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Step 1.4.1

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

$x = \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.4.2

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

$x = - \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

Step 1.4.3

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

$x = \sqrt{2 (3 - y^{2})}$

$x = - \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$x = - \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$x = - \sqrt{2 (3 - y^{2})}$

$2 x^{2} - 3 y^{2} = 5$

Step 2

Solve the system $x = \sqrt{2 (3 - y^{2})}, 2 x^{2} - 3 y^{2} = 5$ .

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $x$ in $2 x^{2} - 3 y^{2} = 5$ with $\sqrt{2 (3 - y^{2})}$ .

$2 {(\sqrt{2 (3 - y^{2})})}^{2} - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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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 ${\sqrt{2 (3 - y^{2})}}^{2}$ as $2 (3 - y^{2})$ .

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Step 2.1.2.1.1.1.1

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

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.1.2

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

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.1.3

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

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

$x = \sqrt{2 (3 - y^{2})}$

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.

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.1.5

Simplify.

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.2

Apply the distributive property.

$2 (2 \cdot 3 + 2 (- y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.3

Multiply $2$ by $3$ .

$2 (6 + 2 (- y^{2})) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.4

Multiply $- 1$ by $2$ .

$2 (6 - 2 y^{2}) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.5

Apply the distributive property.

$2 \cdot 6 + 2 (- 2 y^{2}) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.6

Multiply $2$ by $6$ .

$12 + 2 (- 2 y^{2}) - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.1.7

Multiply $- 2$ by $2$ .

$12 - 4 y^{2} - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$12 - 4 y^{2} - 3 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.1.2.1.2

Subtract $3 y^{2}$ from $- 4 y^{2}$ .

$12 - 7 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2

Solve for $y$ in $12 - 7 y^{2} = 5$ .

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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 $12$ from both sides of the equation.

$- 7 y^{2} = 5 - 12$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.1.2

Subtract $12$ from $5$ .

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

$x = \sqrt{2 (3 - y^{2})}$

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.2

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

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Step 2.2.2.1

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

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.2.2.1.2

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

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

$x = \sqrt{2 (3 - y^{2})}$

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

$x = \sqrt{2 (3 - y^{2})}$

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

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 7$ by $- 7$ .

$y^{2} = 1$

$x = \sqrt{2 (3 - y^{2})}$

$y^{2} = 1$

$x = \sqrt{2 (3 - y^{2})}$

$y^{2} = 1$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.3

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

$y = \pm \sqrt{1}$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.4

Any root of $1$ is $1$ .

$y = \pm 1$

$x = \sqrt{2 (3 - y^{2})}$

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.

$y = 1$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.5.2

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

$y = - 1$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.2.5.3

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

$y = 1, - 1$

$x = \sqrt{2 (3 - y^{2})}$

$y = 1, - 1$

$x = \sqrt{2 (3 - y^{2})}$

$y = 1, - 1$

$x = \sqrt{2 (3 - y^{2})}$

Step 2.3

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

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Step 2.3.1

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

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

$y = 1$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{2 (3 - {(1)}^{2})}$ .

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Step 2.3.2.1.1

One to any power is one.

$x = \sqrt{2 (3 - 1 \cdot 1)}$

$y = 1$

Step 2.3.2.1.2

Multiply $- 1$ by $1$ .

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

$y = 1$

Step 2.3.2.1.3

Subtract $1$ from $3$ .

$x = \sqrt{2 \cdot 2}$

$y = 1$

Step 2.3.2.1.4

Multiply $2$ by $2$ .

$x = \sqrt{4}$

$y = 1$

Step 2.3.2.1.5

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

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

$y = 1$

Step 2.3.2.1.6

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

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

$x = 2$

$y = 1$

Step 2.4

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 2.4.1

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

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

$y = - 1$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{2 (3 - {(- 1)}^{2})}$ .

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Step 2.4.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.4.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 2.4.2.1.1.1.1

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

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

$y = - 1$

Step 2.4.2.1.1.1.2

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

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

$y = - 1$

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

$y = - 1$

Step 2.4.2.1.1.2

Add $1$ and $2$ .

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

$y = - 1$

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

$y = - 1$

Step 2.4.2.1.2

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

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

$y = - 1$

Step 2.4.2.1.3

Subtract $1$ from $3$ .

$x = \sqrt{2 \cdot 2}$

$y = - 1$

Step 2.4.2.1.4

Multiply $2$ by $2$ .

$x = \sqrt{4}$

$y = - 1$

Step 2.4.2.1.5

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

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

$y = - 1$

Step 2.4.2.1.6

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

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

Step 3

Solve the system $x = - \sqrt{2 (3 - y^{2})}, 2 x^{2} - 3 y^{2} = 5$ .

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Step 3.1

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

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Step 3.1.1

Replace all occurrences of $x$ in $2 x^{2} - 3 y^{2} = 5$ with $- \sqrt{2 (3 - y^{2})}$ .

$2 {(- \sqrt{2 (3 - y^{2})})}^{2} - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

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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 $- \sqrt{2 (3 - y^{2})}$ .

$2 ({(- 1)}^{2} {\sqrt{2 (3 - y^{2})}}^{2}) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.2

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

$2 (1 {\sqrt{2 (3 - y^{2})}}^{2}) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{2 (3 - y^{2})}}^{2}$ by $1$ .

$2 {\sqrt{2 (3 - y^{2})}}^{2} - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{2 (3 - y^{2})}}^{2}$ as $2 (3 - y^{2})$ .

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Step 3.1.2.1.1.4.1

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

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.4.2

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

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.4.3

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

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

$x = - \sqrt{2 (3 - y^{2})}$

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.

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.4.5

Simplify.

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$2 (2 (3 - y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.5

Apply the distributive property.

$2 (2 \cdot 3 + 2 (- y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.6

Multiply $2$ by $3$ .

$2 (6 + 2 (- y^{2})) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.7

Multiply $- 1$ by $2$ .

$2 (6 - 2 y^{2}) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.8

Apply the distributive property.

$2 \cdot 6 + 2 (- 2 y^{2}) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.9

Multiply $2$ by $6$ .

$12 + 2 (- 2 y^{2}) - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.1.10

Multiply $- 2$ by $2$ .

$12 - 4 y^{2} - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$12 - 4 y^{2} - 3 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.1.2.1.2

Subtract $3 y^{2}$ from $- 4 y^{2}$ .

$12 - 7 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

$12 - 7 y^{2} = 5$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2

Solve for $y$ in $12 - 7 y^{2} = 5$ .

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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 $12$ from both sides of the equation.

$- 7 y^{2} = 5 - 12$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.1.2

Subtract $12$ from $5$ .

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

$x = - \sqrt{2 (3 - y^{2})}$

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.2

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

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Step 3.2.2.1

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

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.2.2.1.2

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

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

$x = - \sqrt{2 (3 - y^{2})}$

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

$x = - \sqrt{2 (3 - y^{2})}$

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

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 7$ by $- 7$ .

$y^{2} = 1$

$x = - \sqrt{2 (3 - y^{2})}$

$y^{2} = 1$

$x = - \sqrt{2 (3 - y^{2})}$

$y^{2} = 1$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.3

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

$y = \pm \sqrt{1}$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.4

Any root of $1$ is $1$ .

$y = \pm 1$

$x = - \sqrt{2 (3 - y^{2})}$

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.

$y = 1$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.5.2

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

$y = - 1$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.2.5.3

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

$y = 1, - 1$

$x = - \sqrt{2 (3 - y^{2})}$

$y = 1, - 1$

$x = - \sqrt{2 (3 - y^{2})}$

$y = 1, - 1$

$x = - \sqrt{2 (3 - y^{2})}$

Step 3.3

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

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Step 3.3.1

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

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

$y = 1$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{2 (3 - {(1)}^{2})}$ .

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Step 3.3.2.1.1

One to any power is one.

$x = - \sqrt{2 (3 - 1 \cdot 1)}$

$y = 1$

Step 3.3.2.1.2

Multiply $- 1$ by $1$ .

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

$y = 1$

Step 3.3.2.1.3

Subtract $1$ from $3$ .

$x = - \sqrt{2 \cdot 2}$

$y = 1$

Step 3.3.2.1.4

Multiply $2$ by $2$ .

$x = - \sqrt{4}$

$y = 1$

Step 3.3.2.1.5

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

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

$y = 1$

Step 3.3.2.1.6

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

$x = - 1 \cdot 2$

$y = 1$

Step 3.3.2.1.7

Multiply $- 1$ by $2$ .

$x = - 2$

$y = 1$

$x = - 2$

$y = 1$

$x = - 2$

$y = 1$

$x = - 2$

$y = 1$

Step 3.4

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 3.4.1

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

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

$y = - 1$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{2 (3 - {(- 1)}^{2})}$ .

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Step 3.4.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.4.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.4.2.1.1.1.1

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

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

$y = - 1$

Step 3.4.2.1.1.1.2

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

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

$y = - 1$

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

$y = - 1$

Step 3.4.2.1.1.2

Add $1$ and $2$ .

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

$y = - 1$

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

$y = - 1$

Step 3.4.2.1.2

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

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

$y = - 1$

Step 3.4.2.1.3

Subtract $1$ from $3$ .

$x = - \sqrt{2 \cdot 2}$

$y = - 1$

Step 3.4.2.1.4

Multiply $2$ by $2$ .

$x = - \sqrt{4}$

$y = - 1$

Step 3.4.2.1.5

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

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

$y = - 1$

Step 3.4.2.1.6

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

$x = - 1 \cdot 2$

$y = - 1$

Step 3.4.2.1.7

Multiply $- 1$ by $2$ .

$x = - 2$

$y = - 1$

$x = - 2$

$y = - 1$

$x = - 2$

$y = - 1$

$x = - 2$

$y = - 1$

$x = - 2$

$y = - 1$

Step 4

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

$(2, 1)$

$(2, - 1)$

$(- 2, 1)$

$(- 2, - 1)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, 1), (2, - 1), (- 2, 1), (- 2, - 1)$

Equation Form:

$x = 2, y = 1$

$x = 2, y = - 1$

$x = - 2, y = 1$

$x = - 2, y = - 1$

Step 6