Finite Math Examples

$2 x^{2} + y^{2} = 18$ , $x y = 4$

Step 1

Divide each term in $x y = 4$ by $y$ and simplify.

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Step 1.1

Divide each term in $x y = 4$ by $y$ .

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

$2 x^{2} + y^{2} = 18$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $y$ .

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Step 1.2.1.1

Cancel the common factor.

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

$2 x^{2} + y^{2} = 18$

Step 1.2.1.2

Divide $x$ by $1$ .

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

$2 x^{2} + y^{2} = 18$

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

$2 x^{2} + y^{2} = 18$

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

$2 x^{2} + y^{2} = 18$

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

$2 x^{2} + y^{2} = 18$

Step 2

Replace all occurrences of $x$ with $\frac{4}{y}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x^{2} + y^{2} = 18$ with $\frac{4}{y}$ .

$2 {(\frac{4}{y})}^{2} + y^{2} = 18$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the product rule to $\frac{4}{y}$ .

$2 (\frac{4^{2}}{y^{2}}) + y^{2} = 18$

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

Step 2.2.1.2

Raise $4$ to the power of $2$ .

$2 (\frac{16}{y^{2}}) + y^{2} = 18$

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

Step 2.2.1.3

Multiply $2 \frac{16}{y^{2}}$ .

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Step 2.2.1.3.1

Combine $2$ and $\frac{16}{y^{2}}$ .

$\frac{2 \cdot 16}{y^{2}} + y^{2} = 18$

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

Step 2.2.1.3.2

Multiply $2$ by $16$ .

$\frac{32}{y^{2}} + y^{2} = 18$

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

$\frac{32}{y^{2}} + y^{2} = 18$

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

$\frac{32}{y^{2}} + y^{2} = 18$

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

$\frac{32}{y^{2}} + y^{2} = 18$

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

$\frac{32}{y^{2}} + y^{2} = 18$

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

Step 3

Solve for $y$ in $\frac{32}{y^{2}} + y^{2} = 18$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2}, 1, 1$

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

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{2}$

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

$y^{2}$

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

Step 3.2

Multiply each term in $\frac{32}{y^{2}} + y^{2} = 18$ by $y^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{32}{y^{2}} + y^{2} = 18$ by $y^{2}$ .

$\frac{32}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 18 y^{2}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $y^{2}$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{32}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 18 y^{2}$

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

Step 3.2.2.1.1.2

Rewrite the expression.

$32 + y^{2} y^{2} = 18 y^{2}$

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

$32 + y^{2} y^{2} = 18 y^{2}$

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

Step 3.2.2.1.2

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

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Step 3.2.2.1.2.1

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

$32 + y^{2 + 2} = 18 y^{2}$

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

Step 3.2.2.1.2.2

Add $2$ and $2$ .

$32 + y^{4} = 18 y^{2}$

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

$32 + y^{4} = 18 y^{2}$

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

$32 + y^{4} = 18 y^{2}$

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

$32 + y^{4} = 18 y^{2}$

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

$32 + y^{4} = 18 y^{2}$

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

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $18 y^{2}$ from both sides of the equation.

$32 + y^{4} - 18 y^{2} = 0$

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

Step 3.3.2

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 18 u + 32 = 0$

$u = y^{2}$

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

Step 3.3.3

Factor $u^{2} - 18 u + 32$ using the AC method.

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Step 3.3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $- 18$ .

$- 16, - 2$

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

Step 3.3.3.2

Write the factored form using these integers.

$(u - 16) (u - 2) = 0$

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

$(u - 16) (u - 2) = 0$

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

Step 3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 16 = 0$

$u - 2 = 0$

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

Step 3.3.5

Set $u - 16$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

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

Step 3.3.5.2

Add $16$ to both sides of the equation.

$u = 16$

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

$u = 16$

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

Step 3.3.6

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 3.3.6.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

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

Step 3.3.6.2

Add $2$ to both sides of the equation.

$u = 2$

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

$u = 2$

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

Step 3.3.7

The final solution is all the values that make $(u - 16) (u - 2) = 0$ true.

$u = 16, 2$

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

Step 3.3.8

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 16$

$y^{2} = 2$

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

Step 3.3.9

Solve the first equation for $y$ .

$y^{2} = 16$

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

Step 3.3.10

Solve the equation for $y$ .

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Step 3.3.10.1

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

$y = \pm \sqrt{16}$

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

Step 3.3.10.2

Simplify $\pm \sqrt{16}$ .

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Step 3.3.10.2.1

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

$y = \pm \sqrt{4^{2}}$

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

Step 3.3.10.2.2

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

$y = \pm 4$

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

$y = \pm 4$

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

Step 3.3.10.3

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

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Step 3.3.10.3.1

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

$y = 4$

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

Step 3.3.10.3.2

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

$y = - 4$

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

Step 3.3.10.3.3

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

$y = 4, - 4$

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

$y = 4, - 4$

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

$y = 4, - 4$

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

Step 3.3.11

Solve the second equation for $y$ .

$y^{2} = 2$

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

Step 3.3.12

Solve the equation for $y$ .

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Step 3.3.12.1

Remove parentheses.

$y^{2} = 2$

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

Step 3.3.12.2

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

$y = \pm \sqrt{2}$

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

Step 3.3.12.3

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

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Step 3.3.12.3.1

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

$y = \sqrt{2}$

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

Step 3.3.12.3.2

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

$y = - \sqrt{2}$

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

Step 3.3.12.3.3

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

$y = \sqrt{2}, - \sqrt{2}$

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

$y = \sqrt{2}, - \sqrt{2}$

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

$y = \sqrt{2}, - \sqrt{2}$

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

Step 3.3.13

The solution to $y^{4} - 18 y^{2} + 32 = 0$ is $y = 4, - 4, \sqrt{2}, - \sqrt{2}$ .

$y = 4, - 4, \sqrt{2}, - \sqrt{2}$

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

$y = 4, - 4, \sqrt{2}, - \sqrt{2}$

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

$y = 4, - 4, \sqrt{2}, - \sqrt{2}$

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

Step 4

Replace all occurrences of $y$ with $4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $4$ .

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

$y = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $4$ by $4$ .

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

Step 5

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $- 4$ .

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

$y = - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Divide $4$ by $- 4$ .

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

Step 6

Replace all occurrences of $y$ with $4$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $4$ .

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

$y = 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Divide $4$ by $4$ .

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

Step 7

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $- 4$ .

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

$y = - 4$

Step 7.2

Simplify the right side.

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Step 7.2.1

Divide $4$ by $- 4$ .

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

Step 8

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

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Step 8.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $\sqrt{2}$ .

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

$y = \sqrt{2}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{4}{\sqrt{2}}$ .

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Step 8.2.1.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

$y = \sqrt{2}$

Step 8.2.1.2

Combine and simplify the denominator.

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Step 8.2.1.2.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

$y = \sqrt{2}$

Step 8.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = \sqrt{2}$

Step 8.2.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = \sqrt{2}$

Step 8.2.1.2.4

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

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$y = \sqrt{2}$

Step 8.2.1.2.5

Add $1$ and $1$ .

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

$y = \sqrt{2}$

Step 8.2.1.2.6

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

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Step 8.2.1.2.6.1

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

$x = \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$y = \sqrt{2}$

Step 8.2.1.2.6.2

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

$x = \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{2}$

Step 8.2.1.2.6.3

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

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

$y = \sqrt{2}$

Step 8.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 8.2.1.2.6.4.1

Cancel the common factor.

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

$y = \sqrt{2}$

Step 8.2.1.2.6.4.2

Rewrite the expression.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Step 8.2.1.2.6.5

Evaluate the exponent.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Step 8.2.1.3

Cancel the common factor of $4$ and $2$ .

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Step 8.2.1.3.1

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2}$

$y = \sqrt{2}$

Step 8.2.1.3.2

Cancel the common factors.

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Step 8.2.1.3.2.1

Factor $2$ out of $2$ .

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

$y = \sqrt{2}$

Step 8.2.1.3.2.2

Cancel the common factor.

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

$y = \sqrt{2}$

Step 8.2.1.3.2.3

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1}$

$y = \sqrt{2}$

Step 8.2.1.3.2.4

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

Step 9

Replace all occurrences of $y$ with $4$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $4$ .

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

$y = 4$

Step 9.2

Simplify the right side.

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Step 9.2.1

Divide $4$ by $4$ .

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

Step 10

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $- 4$ .

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

$y = - 4$

Step 10.2

Simplify the right side.

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Step 10.2.1

Divide $4$ by $- 4$ .

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

$x = - 1$

$y = - 4$

Step 11

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

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Step 11.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $\sqrt{2}$ .

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

$y = \sqrt{2}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{4}{\sqrt{2}}$ .

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Step 11.2.1.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

$y = \sqrt{2}$

Step 11.2.1.2

Combine and simplify the denominator.

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Step 11.2.1.2.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

$y = \sqrt{2}$

Step 11.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = \sqrt{2}$

Step 11.2.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = \sqrt{2}$

Step 11.2.1.2.4

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

$x = \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$y = \sqrt{2}$

Step 11.2.1.2.5

Add $1$ and $1$ .

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

$y = \sqrt{2}$

Step 11.2.1.2.6

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

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Step 11.2.1.2.6.1

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

$x = \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$y = \sqrt{2}$

Step 11.2.1.2.6.2

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

$x = \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{2}$

Step 11.2.1.2.6.3

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

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

$y = \sqrt{2}$

Step 11.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.2.6.4.1

Cancel the common factor.

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

$y = \sqrt{2}$

Step 11.2.1.2.6.4.2

Rewrite the expression.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Step 11.2.1.2.6.5

Evaluate the exponent.

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

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

$y = \sqrt{2}$

Step 11.2.1.3

Cancel the common factor of $4$ and $2$ .

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Step 11.2.1.3.1

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2}$

$y = \sqrt{2}$

Step 11.2.1.3.2

Cancel the common factors.

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Step 11.2.1.3.2.1

Factor $2$ out of $2$ .

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

$y = \sqrt{2}$

Step 11.2.1.3.2.2

Cancel the common factor.

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

$y = \sqrt{2}$

Step 11.2.1.3.2.3

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1}$

$y = \sqrt{2}$

Step 11.2.1.3.2.4

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

$x = 2 \sqrt{2}$

$y = \sqrt{2}$

Step 12

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

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Step 12.1

Replace all occurrences of $y$ in $x = \frac{4}{y}$ with $- \sqrt{2}$ .

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

$y = - \sqrt{2}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{4}{- \sqrt{2}}$ .

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Step 12.2.1.1

Move the negative in front of the fraction.

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

$y = - \sqrt{2}$

Step 12.2.1.2

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = - (\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

$y = - \sqrt{2}$

Step 12.2.1.3

Combine and simplify the denominator.

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Step 12.2.1.3.1

Multiply $\frac{4}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

$y = - \sqrt{2}$

Step 12.2.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = - \sqrt{2}$

Step 12.2.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

$y = - \sqrt{2}$

Step 12.2.1.3.4

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

$x = - \frac{4 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

$y = - \sqrt{2}$

Step 12.2.1.3.5

Add $1$ and $1$ .

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

$y = - \sqrt{2}$

Step 12.2.1.3.6

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

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Step 12.2.1.3.6.1

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

$x = - \frac{4 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{2}$

Step 12.2.1.3.6.2

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

$x = - \frac{4 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{2}$

Step 12.2.1.3.6.3

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

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

$y = - \sqrt{2}$

Step 12.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 12.2.1.3.6.4.1

Cancel the common factor.

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

$y = - \sqrt{2}$

Step 12.2.1.3.6.4.2

Rewrite the expression.

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

Step 12.2.1.3.6.5

Evaluate the exponent.

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

Step 12.2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 12.2.1.4.1

Factor $2$ out of $4 \sqrt{2}$ .

$x = - \frac{2 (2 \sqrt{2})}{2}$

$y = - \sqrt{2}$

Step 12.2.1.4.2

Cancel the common factors.

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Step 12.2.1.4.2.1

Factor $2$ out of $2$ .

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

$y = - \sqrt{2}$

Step 12.2.1.4.2.2

Cancel the common factor.

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

$y = - \sqrt{2}$

Step 12.2.1.4.2.3

Rewrite the expression.

$x = - \frac{2 \sqrt{2}}{1}$

$y = - \sqrt{2}$

Step 12.2.1.4.2.4

Divide $2 \sqrt{2}$ by $1$ .

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

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

$y = - \sqrt{2}$

Step 12.2.1.5

Multiply $2$ by $- 1$ .

$x = - 2 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - \sqrt{2}$

$x = - 2 \sqrt{2}$

$y = - \sqrt{2}$

Step 13

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

$(1, 4)$

$(- 1, - 4)$

$(2 \sqrt{2}, \sqrt{2})$

$(- 2 \sqrt{2}, - \sqrt{2})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(1, 4), (- 1, - 4), (2 \sqrt{2}, \sqrt{2}), (- 2 \sqrt{2}, - \sqrt{2})$

Equation Form:

$x = 1, y = 4$

$x = - 1, y = - 4$

$x = 2 \sqrt{2}, y = \sqrt{2}$

$x = - 2 \sqrt{2}, y = - \sqrt{2}$

Step 15