Precalculus Examples

$x^{2} + y^{2} = 50$ , $x y = 156$

Step 1

Divide each term in $x y = 156$ by $y$ and simplify.

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Step 1.1

Divide each term in $x y = 156$ by $y$ .

$\frac{x y}{y} = \frac{156}{y}$

$x^{2} + y^{2} = 50$

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.

$\frac{x y}{y} = \frac{156}{y}$

$x^{2} + y^{2} = 50$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{156}{y}$

$x^{2} + y^{2} = 50$

$x = \frac{156}{y}$

$x^{2} + y^{2} = 50$

$x = \frac{156}{y}$

$x^{2} + y^{2} = 50$

$x = \frac{156}{y}$

$x^{2} + y^{2} = 50$

Step 2

Replace all occurrences of $x$ with $\frac{156}{y}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 50$ with $\frac{156}{y}$ .

${(\frac{156}{y})}^{2} + y^{2} = 50$

$x = \frac{156}{y}$

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 $\frac{156}{y}$ .

$\frac{156^{2}}{y^{2}} + y^{2} = 50$

$x = \frac{156}{y}$

Step 2.2.1.2

Raise $156$ to the power of $2$ .

$\frac{24336}{y^{2}} + y^{2} = 50$

$x = \frac{156}{y}$

$\frac{24336}{y^{2}} + y^{2} = 50$

$x = \frac{156}{y}$

$\frac{24336}{y^{2}} + y^{2} = 50$

$x = \frac{156}{y}$

$\frac{24336}{y^{2}} + y^{2} = 50$

$x = \frac{156}{y}$

Step 3

Solve for $y$ in $\frac{24336}{y^{2}} + y^{2} = 50$ .

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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.

$y^{2}, 1, 1$

$x = \frac{156}{y}$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{2}$

$x = \frac{156}{y}$

$y^{2}$

$x = \frac{156}{y}$

Step 3.2

Multiply each term in $\frac{24336}{y^{2}} + y^{2} = 50$ by $y^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{24336}{y^{2}} + y^{2} = 50$ by $y^{2}$ .

$\frac{24336}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 50 y^{2}$

$x = \frac{156}{y}$

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.

$\frac{24336}{y^{2}} \cdot y^{2} + y^{2} y^{2} = 50 y^{2}$

$x = \frac{156}{y}$

Step 3.2.2.1.1.2

Rewrite the expression.

$24336 + y^{2} y^{2} = 50 y^{2}$

$x = \frac{156}{y}$

$24336 + y^{2} y^{2} = 50 y^{2}$

$x = \frac{156}{y}$

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.

$24336 + y^{2 + 2} = 50 y^{2}$

$x = \frac{156}{y}$

Step 3.2.2.1.2.2

Add $2$ and $2$ .

$24336 + y^{4} = 50 y^{2}$

$x = \frac{156}{y}$

$24336 + y^{4} = 50 y^{2}$

$x = \frac{156}{y}$

$24336 + y^{4} = 50 y^{2}$

$x = \frac{156}{y}$

$24336 + y^{4} = 50 y^{2}$

$x = \frac{156}{y}$

$24336 + y^{4} = 50 y^{2}$

$x = \frac{156}{y}$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $50 y^{2}$ from both sides of the equation.

$24336 + y^{4} - 50 y^{2} = 0$

$x = \frac{156}{y}$

Step 3.3.2

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

$u^{2} - 50 u + 24336 = 0$

$u = y^{2}$

$x = \frac{156}{y}$

Step 3.3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = \frac{156}{y}$

Step 3.3.4

Substitute the values $a = 1$ , $b = - 50$ , and $c = 24336$ into the quadratic formula and solve for $u$ .

$\frac{50 \pm \sqrt{{(- 50)}^{2} - 4 \cdot (1 \cdot 24336)}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Simplify the numerator.

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Step 3.3.5.1.1

Raise $- 50$ to the power of $2$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 24336$ .

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Step 3.3.5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.2.2

Multiply $- 4$ by $24336$ .

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.3

Subtract $97344$ from $2500$ .

$u = \frac{50 \pm \sqrt{- 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.4

Rewrite $- 94844$ as $- 1 (94844)$ .

$u = \frac{50 \pm \sqrt{- 1 \cdot 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.5

Rewrite $\sqrt{- 1 (94844)}$ as $\sqrt{- 1} \cdot \sqrt{94844}$ .

$u = \frac{50 \pm \sqrt{- 1} \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{50 \pm i \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.7

Rewrite $94844$ as $2^{2} \cdot 23711$ .

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Step 3.3.5.1.7.1

Factor $4$ out of $94844$ .

$u = \frac{50 \pm i \cdot \sqrt{4 (23711)}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.7.2

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

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.8

Pull terms out from under the radical.

$u = \frac{50 \pm i \cdot (2 \sqrt{23711})}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.1.9

Move $2$ to the left of $i$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.5.2

Multiply $2$ by $1$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2}$

$x = \frac{156}{y}$

Step 3.3.5.3

Simplify $\frac{50 \pm 2 i \sqrt{23711}}{2}$ .

$u = 25 \pm i \sqrt{23711}$

$x = \frac{156}{y}$

$u = 25 \pm i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

Raise $- 50$ to the power of $2$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.2

Multiply $- 4 \cdot 1 \cdot 24336$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.2.2

Multiply $- 4$ by $24336$ .

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.3

Subtract $97344$ from $2500$ .

$u = \frac{50 \pm \sqrt{- 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.4

Rewrite $- 94844$ as $- 1 (94844)$ .

$u = \frac{50 \pm \sqrt{- 1 \cdot 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.5

Rewrite $\sqrt{- 1 (94844)}$ as $\sqrt{- 1} \cdot \sqrt{94844}$ .

$u = \frac{50 \pm \sqrt{- 1} \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{50 \pm i \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.7

Rewrite $94844$ as $2^{2} \cdot 23711$ .

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Step 3.3.6.1.7.1

Factor $4$ out of $94844$ .

$u = \frac{50 \pm i \cdot \sqrt{4 (23711)}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.7.2

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

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.8

Pull terms out from under the radical.

$u = \frac{50 \pm i \cdot (2 \sqrt{23711})}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.1.9

Move $2$ to the left of $i$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.6.2

Multiply $2$ by $1$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2}$

$x = \frac{156}{y}$

Step 3.3.6.3

Simplify $\frac{50 \pm 2 i \sqrt{23711}}{2}$ .

$u = 25 \pm i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.6.4

Change the $\pm$ to $+$ .

$u = 25 + i \sqrt{23711}$

$x = \frac{156}{y}$

$u = 25 + i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.3.7.1

Simplify the numerator.

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Step 3.3.7.1.1

Raise $- 50$ to the power of $2$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.2

Multiply $- 4 \cdot 1 \cdot 24336$ .

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Step 3.3.7.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{50 \pm \sqrt{2500 - 4 \cdot 24336}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.2.2

Multiply $- 4$ by $24336$ .

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm \sqrt{2500 - 97344}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.3

Subtract $97344$ from $2500$ .

$u = \frac{50 \pm \sqrt{- 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.4

Rewrite $- 94844$ as $- 1 (94844)$ .

$u = \frac{50 \pm \sqrt{- 1 \cdot 94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.5

Rewrite $\sqrt{- 1 (94844)}$ as $\sqrt{- 1} \cdot \sqrt{94844}$ .

$u = \frac{50 \pm \sqrt{- 1} \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{50 \pm i \cdot \sqrt{94844}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.7

Rewrite $94844$ as $2^{2} \cdot 23711$ .

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Step 3.3.7.1.7.1

Factor $4$ out of $94844$ .

$u = \frac{50 \pm i \cdot \sqrt{4 (23711)}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.7.2

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

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm i \cdot \sqrt{2^{2} \cdot 23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.8

Pull terms out from under the radical.

$u = \frac{50 \pm i \cdot (2 \sqrt{23711})}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.1.9

Move $2$ to the left of $i$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

$u = \frac{50 \pm 2 i \sqrt{23711}}{2 \cdot 1}$

$x = \frac{156}{y}$

Step 3.3.7.2

Multiply $2$ by $1$ .

$u = \frac{50 \pm 2 i \sqrt{23711}}{2}$

$x = \frac{156}{y}$

Step 3.3.7.3

Simplify $\frac{50 \pm 2 i \sqrt{23711}}{2}$ .

$u = 25 \pm i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.7.4

Change the $\pm$ to $-$ .

$u = 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

$u = 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.8

The final answer is the combination of both solutions.

$u = 25 + i \sqrt{23711}, 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.9

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

$y^{2} = 25 + i \sqrt{23711}$

$y^{2} = 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.10

Solve the first equation for $y$ .

$y^{2} = 25 + i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.11

Solve the equation for $y$ .

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Step 3.3.11.1

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

$y = \pm \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.11.2

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

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Step 3.3.11.2.1

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

$y = \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.11.2.2

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

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.11.2.3

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

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.12

Solve the second equation for $y$ .

$y^{2} = 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.13

Solve the equation for $y$ .

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Step 3.3.13.1

Remove parentheses.

$y^{2} = 25 - i \sqrt{23711}$

$x = \frac{156}{y}$

Step 3.3.13.2

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

$y = \pm \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.13.3

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

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Step 3.3.13.3.1

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

$y = \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.13.3.2

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

$y = - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.13.3.3

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

$y = \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 3.3.14

The solution to $y^{4} - 50 y^{2} + 24336 = 0$ is $y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}, \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$ .

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}, \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}, \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

$y = \sqrt{25 + i \sqrt{23711}}, - \sqrt{25 + i \sqrt{23711}}, \sqrt{25 - i \sqrt{23711}}, - \sqrt{25 - i \sqrt{23711}}$

$x = \frac{156}{y}$

Step 4

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $\sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = \sqrt{25 + i \sqrt{23711}}$

Step 5

Replace all occurrences of $y$ with $- \sqrt{25 + i \sqrt{23711}}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $- \sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{- \sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Move the negative in front of the fraction.

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 6

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $\sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = \sqrt{25 + i \sqrt{23711}}$

Step 7

Replace all occurrences of $y$ with $- \sqrt{25 + i \sqrt{23711}}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $- \sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{- \sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Move the negative in front of the fraction.

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 8

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $\sqrt{25 - i \sqrt{23711}}$ .

$x = \frac{156}{\sqrt{25 - i \sqrt{23711}}}$

$y = \sqrt{25 - i \sqrt{23711}}$

Step 9

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $\sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = \sqrt{25 + i \sqrt{23711}}$

Step 10

Replace all occurrences of $y$ with $- \sqrt{25 + i \sqrt{23711}}$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $- \sqrt{25 + i \sqrt{23711}}$ .

$x = \frac{156}{- \sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 10.2

Simplify the right side.

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Step 10.2.1

Move the negative in front of the fraction.

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}$

$y = - \sqrt{25 + i \sqrt{23711}}$

Step 11

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $\sqrt{25 - i \sqrt{23711}}$ .

$x = \frac{156}{\sqrt{25 - i \sqrt{23711}}}$

$y = \sqrt{25 - i \sqrt{23711}}$

Step 12

Replace all occurrences of $y$ with $- \sqrt{25 - i \sqrt{23711}}$ in each equation.

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Step 12.1

Replace all occurrences of $y$ in $x = \frac{156}{y}$ with $- \sqrt{25 - i \sqrt{23711}}$ .

$x = \frac{156}{- \sqrt{25 - i \sqrt{23711}}}$

$y = - \sqrt{25 - i \sqrt{23711}}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Move the negative in front of the fraction.

$x = - \frac{156}{\sqrt{25 - i \sqrt{23711}}}$

$y = - \sqrt{25 - i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 - i \sqrt{23711}}}$

$y = - \sqrt{25 - i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 - i \sqrt{23711}}}$

$y = - \sqrt{25 - i \sqrt{23711}}$

Step 13

List all of the solutions.

$x = \frac{156}{\sqrt{25 + i \sqrt{23711}}}, y = \sqrt{25 + i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 + i \sqrt{23711}}}, y = - \sqrt{25 + i \sqrt{23711}}$

$x = \frac{156}{\sqrt{25 - i \sqrt{23711}}}, y = \sqrt{25 - i \sqrt{23711}}$

$x = - \frac{156}{\sqrt{25 - i \sqrt{23711}}}, y = - \sqrt{25 - i \sqrt{23711}}$

Step 14