Finite Math Examples

$x^{2} + y^{2} = 25$ , $x^{2} + 4 y^{2} = 52$

Step 1

Solve for $x$ in $x^{2} + y^{2} = 25$ .

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Step 1.1

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

$x^{2} = 25 - y^{2}$

$x^{2} + 4 y^{2} = 52$

Step 1.2

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

$x = \pm \sqrt{25 - y^{2}}$

$x^{2} + 4 y^{2} = 52$

Step 1.3

Simplify $\pm \sqrt{25 - y^{2}}$ .

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Step 1.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} - y^{2}}$

$x^{2} + 4 y^{2} = 52$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = y$ .

$x = \pm \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

$x = \pm \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

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.

$x = \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

Step 1.4.2

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

$x = - \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

Step 1.4.3

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

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x^{2} + 4 y^{2} = 52$

Step 2

Solve the system $x = \sqrt{(5 + y) (5 - y)}, x^{2} + 4 y^{2} = 52$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{(5 + y) (5 - y)}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + 4 y^{2} = 52$ with $\sqrt{(5 + y) (5 - y)}$ .

${(\sqrt{(5 + y) (5 - y)})}^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify ${(\sqrt{(5 + y) (5 - y)})}^{2} + 4 y^{2}$ .

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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 ${\sqrt{(5 + y) (5 - y)}}^{2}$ as $(5 + y) (5 - y)$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + y) (5 - y)}$ as ${((5 + y) (5 - y))}^{\frac{1}{2}}$ .

${({((5 + y) (5 - y))}^{\frac{1}{2}})}^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.1.2

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

${((5 + y) (5 - y))}^{\frac{1}{2} \cdot 2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.1.3

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

${((5 + y) (5 - y))}^{\frac{2}{2}} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

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.

${((5 + y) (5 - y))}^{\frac{2}{2}} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$((5 + y) (5 - y)) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$((5 + y) (5 - y)) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.1.5

Simplify.

$(5 + y) (5 - y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$(5 + y) (5 - y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.2

Expand $(5 + y) (5 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$5 (5 - y) + y (5 - y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y (5 - y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$5 \cdot 5 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $5$ by $5$ .

$25 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.1.2

Multiply $- 1$ by $5$ .

$25 - 5 y + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.1.3

Move $5$ to the left of $y$ .

$25 - 5 y + 5 \cdot y + y (- y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$25 - 5 y + 5 y - y \cdot y + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.1.3.1.5.1

Move $y$ .

$25 - 5 y + 5 y - (y \cdot y) + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.2

Add $- 5 y$ and $5 y$ .

$25 + 0 - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.1.3.3

Add $25$ and $0$ .

$25 - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 - y^{2} + 4 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.1.2.1.2

Add $- y^{2}$ and $4 y^{2}$ .

$25 + 3 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2

Solve for $y$ in $25 + 3 y^{2} = 52$ .

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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 $25$ from both sides of the equation.

$3 y^{2} = 52 - 25$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.1.2

Subtract $25$ from $52$ .

$3 y^{2} = 27$

$x = \sqrt{(5 + y) (5 - y)}$

$3 y^{2} = 27$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.2

Divide each term in $3 y^{2} = 27$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y^{2} = 27$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{27}{3}$

$x = \sqrt{(5 + y) (5 - y)}$

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.

$\frac{3 y^{2}}{3} = \frac{27}{3}$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.2.2.1.2

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

$y^{2} = \frac{27}{3}$

$x = \sqrt{(5 + y) (5 - y)}$

$y^{2} = \frac{27}{3}$

$x = \sqrt{(5 + y) (5 - y)}$

$y^{2} = \frac{27}{3}$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $27$ by $3$ .

$y^{2} = 9$

$x = \sqrt{(5 + y) (5 - y)}$

$y^{2} = 9$

$x = \sqrt{(5 + y) (5 - y)}$

$y^{2} = 9$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.3

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

$y = \pm \sqrt{9}$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.4

Simplify $\pm \sqrt{9}$ .

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Step 2.2.4.1

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

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

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.4.2

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

$y = \pm 3$

$x = \sqrt{(5 + y) (5 - y)}$

$y = \pm 3$

$x = \sqrt{(5 + y) (5 - y)}$

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.

$y = 3$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.5.2

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

$y = - 3$

$x = \sqrt{(5 + y) (5 - y)}$

Step 2.2.5.3

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

$y = 3, - 3$

$x = \sqrt{(5 + y) (5 - y)}$

$y = 3, - 3$

$x = \sqrt{(5 + y) (5 - y)}$

$y = 3, - 3$

$x = \sqrt{(5 + y) (5 - y)}$

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 $x = \sqrt{(5 + y) (5 - y)}$ with $3$ .

$x = \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

Step 2.3.2

Simplify $x = \sqrt{(5 + 3) (5 - (3))}$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Remove parentheses.

$x = \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

$x = \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $\sqrt{(5 + 3) (5 - (3))}$ .

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Step 2.3.2.2.1.1

Add $5$ and $3$ .

$x = \sqrt{8 (5 - (3))}$

$y = 3$

Step 2.3.2.2.1.2

Multiply $- 1$ by $3$ .

$x = \sqrt{8 (5 - 3)}$

$y = 3$

Step 2.3.2.2.1.3

Subtract $3$ from $5$ .

$x = \sqrt{8 \cdot 2}$

$y = 3$

Step 2.3.2.2.1.4

Multiply $8$ by $2$ .

$x = \sqrt{16}$

$y = 3$

Step 2.3.2.2.1.5

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

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

$y = 3$

Step 2.3.2.2.1.6

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

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

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 $x = \sqrt{(5 + y) (5 - y)}$ with $- 3$ .

$x = \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

Step 2.4.2

Simplify $x = \sqrt{(5 - 3) (5 - (- 3))}$ .

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Remove parentheses.

$x = \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

$x = \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{(5 - 3) (5 - (- 3))}$ .

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Step 2.4.2.2.1.1

Subtract $3$ from $5$ .

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

$y = - 3$

Step 2.4.2.2.1.2

Multiply $- 1$ by $- 3$ .

$x = \sqrt{2 (5 + 3)}$

$y = - 3$

Step 2.4.2.2.1.3

Add $5$ and $3$ .

$x = \sqrt{2 \cdot 8}$

$y = - 3$

Step 2.4.2.2.1.4

Multiply $2$ by $8$ .

$x = \sqrt{16}$

$y = - 3$

Step 2.4.2.2.1.5

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

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

$y = - 3$

Step 2.4.2.2.1.6

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

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

$x = 4$

$y = - 3$

Step 3

Solve the system $x = - \sqrt{(5 + y) (5 - y)}, x^{2} + 4 y^{2} = 52$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{(5 + y) (5 - y)}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + 4 y^{2} = 52$ with $- \sqrt{(5 + y) (5 - y)}$ .

${(- \sqrt{(5 + y) (5 - y)})}^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify ${(- \sqrt{(5 + y) (5 - y)})}^{2} + 4 y^{2}$ .

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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 $- \sqrt{(5 + y) (5 - y)}$ .

${(- 1)}^{2} {\sqrt{(5 + y) (5 - y)}}^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.2

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

$1 {\sqrt{(5 + y) (5 - y)}}^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{(5 + y) (5 - y)}}^{2}$ by $1$ .

${\sqrt{(5 + y) (5 - y)}}^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{(5 + y) (5 - y)}}^{2}$ as $(5 + y) (5 - y)$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + y) (5 - y)}$ as ${((5 + y) (5 - y))}^{\frac{1}{2}}$ .

${({((5 + y) (5 - y))}^{\frac{1}{2}})}^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.4.2

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

${((5 + y) (5 - y))}^{\frac{1}{2} \cdot 2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.4.3

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

${((5 + y) (5 - y))}^{\frac{2}{2}} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

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.

${((5 + y) (5 - y))}^{\frac{2}{2}} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$((5 + y) (5 - y)) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$((5 + y) (5 - y)) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.4.5

Simplify.

$(5 + y) (5 - y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$(5 + y) (5 - y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.5

Expand $(5 + y) (5 - y)$ using the FOIL Method.

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Step 3.1.2.1.1.5.1

Apply the distributive property.

$5 (5 - y) + y (5 - y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.5.2

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y (5 - y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.5.3

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$5 \cdot 5 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6

Simplify and combine like terms.

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Step 3.1.2.1.1.6.1

Simplify each term.

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Step 3.1.2.1.1.6.1.1

Multiply $5$ by $5$ .

$25 + 5 (- y) + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.1.2

Multiply $- 1$ by $5$ .

$25 - 5 y + y \cdot 5 + y (- y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.1.3

Move $5$ to the left of $y$ .

$25 - 5 y + 5 \cdot y + y (- y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.1.4

Rewrite using the commutative property of multiplication.

$25 - 5 y + 5 y - y \cdot y + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.1.6.1.5.1

Move $y$ .

$25 - 5 y + 5 y - (y \cdot y) + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.1.5.2

Multiply $y$ by $y$ .

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 - 5 y + 5 y - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.2

Add $- 5 y$ and $5 y$ .

$25 + 0 - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.1.6.3

Add $25$ and $0$ .

$25 - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 - y^{2} + 4 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.1.2.1.2

Add $- y^{2}$ and $4 y^{2}$ .

$25 + 3 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

$25 + 3 y^{2} = 52$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2

Solve for $y$ in $25 + 3 y^{2} = 52$ .

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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 $25$ from both sides of the equation.

$3 y^{2} = 52 - 25$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.1.2

Subtract $25$ from $52$ .

$3 y^{2} = 27$

$x = - \sqrt{(5 + y) (5 - y)}$

$3 y^{2} = 27$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.2

Divide each term in $3 y^{2} = 27$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 y^{2} = 27$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{27}{3}$

$x = - \sqrt{(5 + y) (5 - y)}$

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.

$\frac{3 y^{2}}{3} = \frac{27}{3}$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.2.2.1.2

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

$y^{2} = \frac{27}{3}$

$x = - \sqrt{(5 + y) (5 - y)}$

$y^{2} = \frac{27}{3}$

$x = - \sqrt{(5 + y) (5 - y)}$

$y^{2} = \frac{27}{3}$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $27$ by $3$ .

$y^{2} = 9$

$x = - \sqrt{(5 + y) (5 - y)}$

$y^{2} = 9$

$x = - \sqrt{(5 + y) (5 - y)}$

$y^{2} = 9$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.3

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

$y = \pm \sqrt{9}$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.4

Simplify $\pm \sqrt{9}$ .

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Step 3.2.4.1

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

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

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.4.2

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

$y = \pm 3$

$x = - \sqrt{(5 + y) (5 - y)}$

$y = \pm 3$

$x = - \sqrt{(5 + y) (5 - y)}$

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.

$y = 3$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.5.2

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

$y = - 3$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.2.5.3

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

$y = 3, - 3$

$x = - \sqrt{(5 + y) (5 - y)}$

$y = 3, - 3$

$x = - \sqrt{(5 + y) (5 - y)}$

$y = 3, - 3$

$x = - \sqrt{(5 + y) (5 - y)}$

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 $x = - \sqrt{(5 + y) (5 - y)}$ with $3$ .

$x = - \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

Step 3.3.2

Simplify $x = - \sqrt{(5 + 3) (5 - (3))}$ .

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Remove parentheses.

$x = - \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

$x = - \sqrt{(5 + 3) (5 - (3))}$

$y = 3$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $- \sqrt{(5 + 3) (5 - (3))}$ .

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Step 3.3.2.2.1.1

Add $5$ and $3$ .

$x = - \sqrt{8 (5 - (3))}$

$y = 3$

Step 3.3.2.2.1.2

Multiply $- 1$ by $3$ .

$x = - \sqrt{8 (5 - 3)}$

$y = 3$

Step 3.3.2.2.1.3

Subtract $3$ from $5$ .

$x = - \sqrt{8 \cdot 2}$

$y = 3$

Step 3.3.2.2.1.4

Multiply $8$ by $2$ .

$x = - \sqrt{16}$

$y = 3$

Step 3.3.2.2.1.5

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

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

$y = 3$

Step 3.3.2.2.1.6

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

$x = - 1 \cdot 4$

$y = 3$

Step 3.3.2.2.1.7

Multiply $- 1$ by $4$ .

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

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 $x = - \sqrt{(5 + y) (5 - y)}$ with $- 3$ .

$x = - \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

Step 3.4.2

Simplify $x = - \sqrt{(5 - 3) (5 - (- 3))}$ .

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Remove parentheses.

$x = - \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

$x = - \sqrt{(5 - 3) (5 - (- 3))}$

$y = - 3$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Simplify $- \sqrt{(5 - 3) (5 - (- 3))}$ .

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Step 3.4.2.2.1.1

Subtract $3$ from $5$ .

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

$y = - 3$

Step 3.4.2.2.1.2

Multiply $- 1$ by $- 3$ .

$x = - \sqrt{2 (5 + 3)}$

$y = - 3$

Step 3.4.2.2.1.3

Add $5$ and $3$ .

$x = - \sqrt{2 \cdot 8}$

$y = - 3$

Step 3.4.2.2.1.4

Multiply $2$ by $8$ .

$x = - \sqrt{16}$

$y = - 3$

Step 3.4.2.2.1.5

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

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

$y = - 3$

Step 3.4.2.2.1.6

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

$x = - 1 \cdot 4$

$y = - 3$

Step 3.4.2.2.1.7

Multiply $- 1$ by $4$ .

$x = - 4$

$y = - 3$

$x = - 4$

$y = - 3$

$x = - 4$

$y = - 3$

$x = - 4$

$y = - 3$

$x = - 4$

$y = - 3$

$x = - 4$

$y = - 3$

Step 4

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

$(4, 3)$

$(4, - 3)$

$(- 4, 3)$

$(- 4, - 3)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(4, 3), (4, - 3), (- 4, 3), (- 4, - 3)$

Equation Form:

$x = 4, y = 3$

$x = 4, y = - 3$

$x = - 4, y = 3$

$x = - 4, y = - 3$

Step 6