Precalculus Examples

$x y = 11$ , $2 y^{2} - x^{2} = 11$

Step 1

Divide each term in $x y = 11$ by $y$ and simplify.

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Step 1.1

Divide each term in $x y = 11$ by $y$ .

$\frac{x y}{y} = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

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.

$\frac{x y}{y} = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

$x = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

$x = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

$x = \frac{11}{y}$

$2 y^{2} - x^{2} = 11$

Step 2

Replace all occurrences of $x$ with $\frac{11}{y}$ in each equation.

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Step 2.1

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

$2 y^{2} - {(\frac{11}{y})}^{2} = 11$

$x = \frac{11}{y}$

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 $\frac{11}{y}$ .

$2 y^{2} - \frac{11^{2}}{y^{2}} = 11$

$x = \frac{11}{y}$

Step 2.2.1.2

Raise $11$ to the power of $2$ .

$2 y^{2} - \frac{121}{y^{2}} = 11$

$x = \frac{11}{y}$

$2 y^{2} - \frac{121}{y^{2}} = 11$

$x = \frac{11}{y}$

$2 y^{2} - \frac{121}{y^{2}} = 11$

$x = \frac{11}{y}$

$2 y^{2} - \frac{121}{y^{2}} = 11$

$x = \frac{11}{y}$

Step 3

Solve for $y$ in $2 y^{2} - \frac{121}{y^{2}} = 11$ .

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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.

$1, y^{2}, 1$

$x = \frac{11}{y}$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{2}$

$x = \frac{11}{y}$

$y^{2}$

$x = \frac{11}{y}$

Step 3.2

Multiply each term in $2 y^{2} - \frac{121}{y^{2}} = 11$ by $y^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $2 y^{2} - \frac{121}{y^{2}} = 11$ by $y^{2}$ .

$2 y^{2} y^{2} - \frac{121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

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

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

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Step 3.2.2.1.1.1

Move $y^{2}$ .

$2 (y^{2} y^{2}) - \frac{121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.2.2.1.1.2

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

$2 y^{2 + 2} - \frac{121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.2.2.1.1.3

Add $2$ and $2$ .

$2 y^{4} - \frac{121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

$2 y^{4} - \frac{121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.2.2.1.2

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

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Step 3.2.2.1.2.1

Move the leading negative in $- \frac{121}{y^{2}}$ into the numerator.

$2 y^{4} + \frac{- 121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.2.2.1.2.2

Cancel the common factor.

$2 y^{4} + \frac{- 121}{y^{2}} \cdot y^{2} = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.2.2.1.2.3

Rewrite the expression.

$2 y^{4} - 121 = 11 y^{2}$

$x = \frac{11}{y}$

$2 y^{4} - 121 = 11 y^{2}$

$x = \frac{11}{y}$

$2 y^{4} - 121 = 11 y^{2}$

$x = \frac{11}{y}$

$2 y^{4} - 121 = 11 y^{2}$

$x = \frac{11}{y}$

$2 y^{4} - 121 = 11 y^{2}$

$x = \frac{11}{y}$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $11 y^{2}$ from both sides of the equation.

$2 y^{4} - 121 - 11 y^{2} = 0$

$x = \frac{11}{y}$

Step 3.3.2

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

$2 u^{2} - 11 u - 121 = 0$

$u = y^{2}$

$x = \frac{11}{y}$

Step 3.3.3

Factor by grouping.

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Step 3.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 121 = - 242$ and whose sum is $b = - 11$ .

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Step 3.3.3.1.1

Factor $- 11$ out of $- 11 u$ .

$2 u^{2} - 11 u - 121 = 0$

$x = \frac{11}{y}$

Step 3.3.3.1.2

Rewrite $- 11$ as $11$ plus $- 22$

$2 u^{2} + (11 - 22) u - 121 = 0$

$x = \frac{11}{y}$

Step 3.3.3.1.3

Apply the distributive property.

$2 u^{2} + 11 u - 22 u - 121 = 0$

$x = \frac{11}{y}$

$2 u^{2} + 11 u - 22 u - 121 = 0$

$x = \frac{11}{y}$

Step 3.3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.3.2.1

Group the first two terms and the last two terms.

$(2 u^{2} + 11 u) - 22 u - 121 = 0$

$x = \frac{11}{y}$

Step 3.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u + 11) - 11 (2 u + 11) = 0$

$x = \frac{11}{y}$

$u (2 u + 11) - 11 (2 u + 11) = 0$

$x = \frac{11}{y}$

Step 3.3.3.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 11$ .

$(2 u + 11) (u - 11) = 0$

$x = \frac{11}{y}$

$(2 u + 11) (u - 11) = 0$

$x = \frac{11}{y}$

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 $0$ .

$2 u + 11 = 0$

$u - 11 = 0$

$x = \frac{11}{y}$

Step 3.3.5

Set $2 u + 11$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $2 u + 11$ equal to $0$ .

$2 u + 11 = 0$

$x = \frac{11}{y}$

Step 3.3.5.2

Solve $2 u + 11 = 0$ for $u$ .

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Step 3.3.5.2.1

Subtract $11$ from both sides of the equation.

$2 u = - 11$

$x = \frac{11}{y}$

Step 3.3.5.2.2

Divide each term in $2 u = - 11$ by $2$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $2 u = - 11$ by $2$ .

$\frac{2 u}{2} = \frac{- 11}{2}$

$x = \frac{11}{y}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 11}{2}$

$x = \frac{11}{y}$

Step 3.3.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 11}{2}$

$x = \frac{11}{y}$

$u = \frac{- 11}{2}$

$x = \frac{11}{y}$

$u = \frac{- 11}{2}$

$x = \frac{11}{y}$

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{11}{2}$

$x = \frac{11}{y}$

$u = - \frac{11}{2}$

$x = \frac{11}{y}$

$u = - \frac{11}{2}$

$x = \frac{11}{y}$

$u = - \frac{11}{2}$

$x = \frac{11}{y}$

$u = - \frac{11}{2}$

$x = \frac{11}{y}$

Step 3.3.6

Set $u - 11$ equal to $0$ and solve for $u$ .

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Step 3.3.6.1

Set $u - 11$ equal to $0$ .

$u - 11 = 0$

$x = \frac{11}{y}$

Step 3.3.6.2

Add $11$ to both sides of the equation.

$u = 11$

$x = \frac{11}{y}$

$u = 11$

$x = \frac{11}{y}$

Step 3.3.7

The final solution is all the values that make $(2 u + 11) (u - 11) = 0$ true.

$u = - \frac{11}{2}, 11$

$x = \frac{11}{y}$

Step 3.3.8

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

$y^{2} = - \frac{11}{2}$

$y^{2} = 11$

$x = \frac{11}{y}$

Step 3.3.9

Solve the first equation for $y$ .

$y^{2} = - \frac{11}{2}$

$x = \frac{11}{y}$

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.

$y = \pm \sqrt{- \frac{11}{2}}$

$x = \frac{11}{y}$

Step 3.3.10.2

Simplify $\pm \sqrt{- \frac{11}{2}}$ .

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Step 3.3.10.2.1

Rewrite $- 1$ as $i^{2}$ .

$y = \pm \sqrt{i^{2} (\frac{11}{2})}$

$x = \frac{11}{y}$

Step 3.3.10.2.2

Pull terms out from under the radical.

$y = \pm i \sqrt{\frac{11}{2}}$

$x = \frac{11}{y}$

Step 3.3.10.2.3

Rewrite $\sqrt{\frac{11}{2}}$ as $\frac{\sqrt{11}}{\sqrt{2}}$ .

$y = \pm i (\frac{\sqrt{11}}{\sqrt{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.4

Multiply $\frac{\sqrt{11}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm i (\frac{\sqrt{11}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5

Combine and simplify the denominator.

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Step 3.3.10.2.5.1

Multiply $\frac{\sqrt{11}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.2

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

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.3

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

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.4

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

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

$x = \frac{11}{y}$

Step 3.3.10.2.5.5

Add $1$ and $1$ .

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{{\sqrt{2}}^{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6

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

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Step 3.3.10.2.5.6.1

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

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6.2

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

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6.3

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

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2^{\frac{2}{2}}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6.4

Cancel the common factor of $2$ .

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Step 3.3.10.2.5.6.4.1

Cancel the common factor.

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2^{\frac{2}{2}}})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6.4.2

Rewrite the expression.

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2})$

$x = \frac{11}{y}$

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2})$

$x = \frac{11}{y}$

Step 3.3.10.2.5.6.5

Evaluate the exponent.

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2})$

$x = \frac{11}{y}$

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2})$

$x = \frac{11}{y}$

$y = \pm i (\frac{\sqrt{11} \sqrt{2}}{2})$

$x = \frac{11}{y}$

Step 3.3.10.2.6

Simplify the numerator.

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Step 3.3.10.2.6.1

Combine using the product rule for radicals.

$y = \pm i (\frac{\sqrt{11 \cdot 2}}{2})$

$x = \frac{11}{y}$

Step 3.3.10.2.6.2

Multiply $11$ by $2$ .

$y = \pm i (\frac{\sqrt{22}}{2})$

$x = \frac{11}{y}$

$y = \pm i (\frac{\sqrt{22}}{2})$

$x = \frac{11}{y}$

Step 3.3.10.2.7

Combine $i$ and $\frac{\sqrt{22}}{2}$ .

$y = \pm \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

$y = \pm \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

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.

$y = \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

Step 3.3.10.3.2

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

$y = - \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

Step 3.3.10.3.3

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

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}$

$x = \frac{11}{y}$

Step 3.3.11

Solve the second equation for $y$ .

$y^{2} = 11$

$x = \frac{11}{y}$

Step 3.3.12

Solve the equation for $y$ .

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Step 3.3.12.1

Remove parentheses.

$y^{2} = 11$

$x = \frac{11}{y}$

Step 3.3.12.2

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

$y = \pm \sqrt{11}$

$x = \frac{11}{y}$

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.

$y = \sqrt{11}$

$x = \frac{11}{y}$

Step 3.3.12.3.2

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

$y = - \sqrt{11}$

$x = \frac{11}{y}$

Step 3.3.12.3.3

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

$y = \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

$y = \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

$y = \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

Step 3.3.13

The solution to $2 y^{4} - 11 y^{2} - 121 = 0$ is $y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}, \sqrt{11}, - \sqrt{11}$ .

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}, \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}, \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

$y = \frac{i \sqrt{22}}{2}, - \frac{i \sqrt{22}}{2}, \sqrt{11}, - \sqrt{11}$

$x = \frac{11}{y}$

Step 4

Replace all occurrences of $y$ with $\frac{i \sqrt{22}}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $\frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{\frac{i \sqrt{22}}{2}}$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{11}{\frac{i \sqrt{22}}{2}}$ .

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Step 4.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (\frac{2}{i \sqrt{22}})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (\frac{2}{4.69041575 i} \cdot \frac{i}{i})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3

Multiply.

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Step 4.2.1.3.1

Combine.

$x = 11 (\frac{2 i}{4.69041575 i i})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2

Simplify the denominator.

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Step 4.2.1.3.2.1

Add parentheses.

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2.2

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2.3

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2.4

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

$x = 11 (\frac{2 i}{4.69041575 i^{1 + 1}})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (\frac{2 i}{4.69041575 i^{2}})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.4

Simplify the expression.

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Step 4.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (\frac{2 i}{- 4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (- \frac{2 (i)}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (- \frac{2 (i)}{4.69041575 (1)})$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.7

Separate fractions.

$x = 11 (- (\frac{2}{4.69041575} \cdot \frac{i}{1}))$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (- (0.42640143 (\frac{i}{1})))$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.9

Divide $i$ by $1$ .

$x = 11 (- (0.42640143 i))$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- 0.42640143 i)$

$y = \frac{i \sqrt{22}}{2}$

Step 4.2.1.11

Multiply $- 0.42640143$ by $11$ .

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

Step 5

Replace all occurrences of $y$ with $- \frac{i \sqrt{22}}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $- \frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{- \frac{i \sqrt{22}}{2}}$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{11}{- \frac{i \sqrt{22}}{2}}$ .

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Step 5.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (- \frac{2}{i \sqrt{22}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (- (\frac{2}{4.69041575 i} \cdot \frac{i}{i}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Combine.

$x = 11 (- \frac{2 i}{4.69041575 i i})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2

Simplify the denominator.

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Step 5.2.1.3.2.1

Add parentheses.

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2.2

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2.3

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2.4

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

$x = 11 (- \frac{2 i}{4.69041575 i^{1 + 1}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (- \frac{2 i}{4.69041575 i^{2}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.4

Simplify the expression.

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Step 5.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (- \frac{2 i}{- 4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (\frac{2 (i)}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (\frac{2 (i)}{4.69041575 (1)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.7

Separate fractions.

$x = 11 (\frac{2}{4.69041575} \cdot \frac{i}{1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (0.42640143 (\frac{i}{1}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.9

Divide $i$ by $1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- (- 0.42640143 i))$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.11

Multiply $- 0.42640143$ by $- 1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 5.2.1.12

Multiply $0.42640143$ by $11$ .

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

Step 6

Replace all occurrences of $y$ with $\frac{i \sqrt{22}}{2}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $\frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{\frac{i \sqrt{22}}{2}}$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{11}{\frac{i \sqrt{22}}{2}}$ .

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Step 6.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (\frac{2}{i \sqrt{22}})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (\frac{2}{4.69041575 i} \cdot \frac{i}{i})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3

Multiply.

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Step 6.2.1.3.1

Combine.

$x = 11 (\frac{2 i}{4.69041575 i i})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2

Simplify the denominator.

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Step 6.2.1.3.2.1

Add parentheses.

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2.2

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2.3

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2.4

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

$x = 11 (\frac{2 i}{4.69041575 i^{1 + 1}})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (\frac{2 i}{4.69041575 i^{2}})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.4

Simplify the expression.

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Step 6.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (\frac{2 i}{- 4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (- \frac{2 (i)}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (- \frac{2 (i)}{4.69041575 (1)})$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.7

Separate fractions.

$x = 11 (- (\frac{2}{4.69041575} \cdot \frac{i}{1}))$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (- (0.42640143 (\frac{i}{1})))$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.9

Divide $i$ by $1$ .

$x = 11 (- (0.42640143 i))$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- 0.42640143 i)$

$y = \frac{i \sqrt{22}}{2}$

Step 6.2.1.11

Multiply $- 0.42640143$ by $11$ .

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

Step 7

Replace all occurrences of $y$ with $- \frac{i \sqrt{22}}{2}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $- \frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{- \frac{i \sqrt{22}}{2}}$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{11}{- \frac{i \sqrt{22}}{2}}$ .

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Step 7.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (- \frac{2}{i \sqrt{22}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (- (\frac{2}{4.69041575 i} \cdot \frac{i}{i}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3

Multiply.

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Step 7.2.1.3.1

Combine.

$x = 11 (- \frac{2 i}{4.69041575 i i})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2

Simplify the denominator.

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Step 7.2.1.3.2.1

Add parentheses.

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2.2

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2.3

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2.4

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

$x = 11 (- \frac{2 i}{4.69041575 i^{1 + 1}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (- \frac{2 i}{4.69041575 i^{2}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.4

Simplify the expression.

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Step 7.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (- \frac{2 i}{- 4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (\frac{2 (i)}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (\frac{2 (i)}{4.69041575 (1)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.7

Separate fractions.

$x = 11 (\frac{2}{4.69041575} \cdot \frac{i}{1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (0.42640143 (\frac{i}{1}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.9

Divide $i$ by $1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- (- 0.42640143 i))$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.11

Multiply $- 0.42640143$ by $- 1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 7.2.1.12

Multiply $0.42640143$ by $11$ .

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

Step 8

Replace all occurrences of $y$ with $\sqrt{11}$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $\sqrt{11}$ .

$x = \frac{11}{\sqrt{11}}$

$y = \sqrt{11}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{11}{\sqrt{11}}$ .

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Step 8.2.1.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \frac{11}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

$y = \sqrt{11}$

Step 8.2.1.2

Combine and simplify the denominator.

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Step 8.2.1.2.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 8.2.1.2.2

Raise $\sqrt{11}$ to the power of $1$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 8.2.1.2.3

Raise $\sqrt{11}$ to the power of $1$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 8.2.1.2.4

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

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

$y = \sqrt{11}$

Step 8.2.1.2.5

Add $1$ and $1$ .

$x = \frac{11 \sqrt{11}}{{\sqrt{11}}^{2}}$

$y = \sqrt{11}$

Step 8.2.1.2.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 8.2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \frac{11 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

$y = \sqrt{11}$

Step 8.2.1.2.6.2

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

$x = \frac{11 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{11}$

Step 8.2.1.2.6.3

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

$x = \frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

$y = \sqrt{11}$

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.

$x = \frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

$y = \sqrt{11}$

Step 8.2.1.2.6.4.2

Rewrite the expression.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 8.2.1.2.6.5

Evaluate the exponent.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 8.2.1.3

Cancel the common factor of $11$ .

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Step 8.2.1.3.1

Cancel the common factor.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 8.2.1.3.2

Divide $\sqrt{11}$ by $1$ .

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

Step 9

Replace all occurrences of $y$ with $\frac{i \sqrt{22}}{2}$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $\frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{\frac{i \sqrt{22}}{2}}$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{11}{\frac{i \sqrt{22}}{2}}$ .

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Step 9.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (\frac{2}{i \sqrt{22}})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (\frac{2}{4.69041575 i} \cdot \frac{i}{i})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3

Multiply.

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Step 9.2.1.3.1

Combine.

$x = 11 (\frac{2 i}{4.69041575 i i})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2

Simplify the denominator.

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Step 9.2.1.3.2.1

Add parentheses.

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2.2

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2.3

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

$x = 11 (\frac{2 i}{4.69041575 (i i)})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2.4

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

$x = 11 (\frac{2 i}{4.69041575 i^{1 + 1}})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (\frac{2 i}{4.69041575 i^{2}})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575 \cdot - 1})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.4

Simplify the expression.

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Step 9.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (\frac{2 i}{- 4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (- \frac{2 (i)}{4.69041575})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (- \frac{2 (i)}{4.69041575 (1)})$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.7

Separate fractions.

$x = 11 (- (\frac{2}{4.69041575} \cdot \frac{i}{1}))$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (- (0.42640143 (\frac{i}{1})))$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.9

Divide $i$ by $1$ .

$x = 11 (- (0.42640143 i))$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- 0.42640143 i)$

$y = \frac{i \sqrt{22}}{2}$

Step 9.2.1.11

Multiply $- 0.42640143$ by $11$ .

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

$x = - 4.69041575 i$

$y = \frac{i \sqrt{22}}{2}$

Step 10

Replace all occurrences of $y$ with $- \frac{i \sqrt{22}}{2}$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $- \frac{i \sqrt{22}}{2}$ .

$x = \frac{11}{- \frac{i \sqrt{22}}{2}}$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $\frac{11}{- \frac{i \sqrt{22}}{2}}$ .

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Step 10.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 11 (- \frac{2}{i \sqrt{22}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.2

Multiply the numerator and denominator of $\frac{2}{4.69041575 i}$ by the conjugate of $4.69041575 i$ to make the denominator real.

$x = 11 (- (\frac{2}{4.69041575 i} \cdot \frac{i}{i}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3

Multiply.

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Step 10.2.1.3.1

Combine.

$x = 11 (- \frac{2 i}{4.69041575 i i})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2

Simplify the denominator.

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Step 10.2.1.3.2.1

Add parentheses.

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2.2

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2.3

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

$x = 11 (- \frac{2 i}{4.69041575 (i i)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2.4

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

$x = 11 (- \frac{2 i}{4.69041575 i^{1 + 1}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2.5

Add $1$ and $1$ .

$x = 11 (- \frac{2 i}{4.69041575 i^{2}})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.3.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (- \frac{2 i}{4.69041575 \cdot - 1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.4

Simplify the expression.

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Step 10.2.1.4.1

Multiply $4.69041575$ by $- 1$ .

$x = 11 (- \frac{2 i}{- 4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.4.2

Move the negative in front of the fraction.

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

$x = 11 (\frac{2 i}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.5

Factor $2$ out of $2 i$ .

$x = 11 (\frac{2 (i)}{4.69041575})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.6

Factor $4.69041575$ out of $4.69041575$ .

$x = 11 (\frac{2 (i)}{4.69041575 (1)})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.7

Separate fractions.

$x = 11 (\frac{2}{4.69041575} \cdot \frac{i}{1})$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.8

Divide $2$ by $4.69041575$ .

$x = 11 (0.42640143 (\frac{i}{1}))$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.9

Divide $i$ by $1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.10

Multiply $0.42640143$ by $- 1$ .

$x = 11 (- (- 0.42640143 i))$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.11

Multiply $- 0.42640143$ by $- 1$ .

$x = 11 (0.42640143 i)$

$y = - \frac{i \sqrt{22}}{2}$

Step 10.2.1.12

Multiply $0.42640143$ by $11$ .

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i$

$y = - \frac{i \sqrt{22}}{2}$

Step 11

Replace all occurrences of $y$ with $\sqrt{11}$ in each equation.

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Step 11.1

Replace all occurrences of $y$ in $x = \frac{11}{y}$ with $\sqrt{11}$ .

$x = \frac{11}{\sqrt{11}}$

$y = \sqrt{11}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{11}{\sqrt{11}}$ .

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Step 11.2.1.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \frac{11}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

$y = \sqrt{11}$

Step 11.2.1.2

Combine and simplify the denominator.

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Step 11.2.1.2.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 11.2.1.2.2

Raise $\sqrt{11}$ to the power of $1$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 11.2.1.2.3

Raise $\sqrt{11}$ to the power of $1$ .

$x = \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = \sqrt{11}$

Step 11.2.1.2.4

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

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

$y = \sqrt{11}$

Step 11.2.1.2.5

Add $1$ and $1$ .

$x = \frac{11 \sqrt{11}}{{\sqrt{11}}^{2}}$

$y = \sqrt{11}$

Step 11.2.1.2.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 11.2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \frac{11 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

$y = \sqrt{11}$

Step 11.2.1.2.6.2

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

$x = \frac{11 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{11}$

Step 11.2.1.2.6.3

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

$x = \frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

$y = \sqrt{11}$

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.

$x = \frac{11 \sqrt{11}}{11^{\frac{2}{2}}}$

$y = \sqrt{11}$

Step 11.2.1.2.6.4.2

Rewrite the expression.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 11.2.1.2.6.5

Evaluate the exponent.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 11.2.1.3

Cancel the common factor of $11$ .

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Step 11.2.1.3.1

Cancel the common factor.

$x = \frac{11 \sqrt{11}}{11}$

$y = \sqrt{11}$

Step 11.2.1.3.2

Divide $\sqrt{11}$ by $1$ .

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

$x = \sqrt{11}$

$y = \sqrt{11}$

Step 12

Replace all occurrences of $y$ with $- \sqrt{11}$ in each equation.

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Step 12.1

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

$x = \frac{11}{- \sqrt{11}}$

$y = - \sqrt{11}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{11}{- \sqrt{11}}$ .

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Step 12.2.1.1

Move the negative in front of the fraction.

$x = - \frac{11}{\sqrt{11}}$

$y = - \sqrt{11}$

Step 12.2.1.2

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = - (\frac{11}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

$y = - \sqrt{11}$

Step 12.2.1.3

Combine and simplify the denominator.

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Step 12.2.1.3.1

Multiply $\frac{11}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = - \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = - \sqrt{11}$

Step 12.2.1.3.2

Raise $\sqrt{11}$ to the power of $1$ .

$x = - \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = - \sqrt{11}$

Step 12.2.1.3.3

Raise $\sqrt{11}$ to the power of $1$ .

$x = - \frac{11 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$y = - \sqrt{11}$

Step 12.2.1.3.4

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

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

$y = - \sqrt{11}$

Step 12.2.1.3.5

Add $1$ and $1$ .

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

$y = - \sqrt{11}$

Step 12.2.1.3.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 12.2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = - \frac{11 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{11}$

Step 12.2.1.3.6.2

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

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

$y = - \sqrt{11}$

Step 12.2.1.3.6.3

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

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

$y = - \sqrt{11}$

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.

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

$y = - \sqrt{11}$

Step 12.2.1.3.6.4.2

Rewrite the expression.

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

Step 12.2.1.3.6.5

Evaluate the exponent.

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

Step 12.2.1.4

Cancel the common factor of $11$ .

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Step 12.2.1.4.1

Cancel the common factor.

$x = - \frac{11 \sqrt{11}}{11}$

$y = - \sqrt{11}$

Step 12.2.1.4.2

Divide $\sqrt{11}$ by $1$ .

$x = - \sqrt{11}$

$y = - \sqrt{11}$

$x = - \sqrt{11}$

$y = - \sqrt{11}$

$x = - \sqrt{11}$

$y = - \sqrt{11}$

$x = - \sqrt{11}$

$y = - \sqrt{11}$

$x = - \sqrt{11}$

$y = - \sqrt{11}$

Step 13

List all of the solutions.

$x = - 4.69041575 i, y = \frac{i \sqrt{22}}{2}$

$x = 4.69041575 i, y = - \frac{i \sqrt{22}}{2}$

$x = \sqrt{11}, y = \sqrt{11}$

$x = - \sqrt{11}, y = - \sqrt{11}$

Step 14