Precalculus Examples

$7 x^{2} - 3 y^{2} + 5 = 0$ , $3 x^{2} + 5 y^{2} = 12$

Step 1

Solve for $x$ in $7 x^{2} - 3 y^{2} + 5 = 0$ .

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Step 1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1

Add $3 y^{2}$ to both sides of the equation.

$7 x^{2} + 5 = 3 y^{2}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.1.2

Subtract $5$ from both sides of the equation.

$7 x^{2} = 3 y^{2} - 5$

$3 x^{2} + 5 y^{2} = 12$

$7 x^{2} = 3 y^{2} - 5$

$3 x^{2} + 5 y^{2} = 12$

Step 1.2

Divide each term in $7 x^{2} = 3 y^{2} - 5$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 x^{2} = 3 y^{2} - 5$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{3 y^{2}}{7} + \frac{- 5}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{3 y^{2}}{7} + \frac{- 5}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.2.2.1.2

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

$x^{2} = \frac{3 y^{2}}{7} + \frac{- 5}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x^{2} = \frac{3 y^{2}}{7} + \frac{- 5}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x^{2} = \frac{3 y^{2}}{7} + \frac{- 5}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x^{2} = \frac{3 y^{2}}{7} - \frac{5}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x^{2} = \frac{3 y^{2}}{7} - \frac{5}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x^{2} = \frac{3 y^{2}}{7} - \frac{5}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.3

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

$x = \pm \sqrt{\frac{3 y^{2}}{7} - \frac{5}{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4

Simplify $\pm \sqrt{\frac{3 y^{2}}{7} - \frac{5}{7}}$ .

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Step 1.4.1

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{3 y^{2} - 5}{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.2

Rewrite $\sqrt{\frac{3 y^{2} - 5}{7}}$ as $\frac{\sqrt{3 y^{2} - 5}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5}}{\sqrt{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.3

Multiply $\frac{\sqrt{3 y^{2} - 5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4

Combine and simplify the denominator.

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Step 1.4.4.1

Multiply $\frac{\sqrt{3 y^{2} - 5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.4

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

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{{\sqrt{7}}^{2}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6.2

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

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6.3

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

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7^{\frac{2}{2}}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7^{\frac{2}{2}}}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \pm \frac{\sqrt{3 y^{2} - 5} \sqrt{7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.5

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(3 y^{2} - 5) \cdot 7}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.4.6

Reorder factors in $\pm \frac{\sqrt{(3 y^{2} - 5) \cdot 7}}{7}$ .

$x = \pm \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \pm \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.5

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

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Step 1.5.1

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

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.5.2

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

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 1.5.3

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

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 x^{2} + 5 y^{2} = 12$

Step 2

Solve the system $x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}, 3 x^{2} + 5 y^{2} = 12$ .

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Step 2.1

Replace all occurrences of $x$ with $\frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $3 x^{2} + 5 y^{2} = 12$ with $\frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ .

$3 {(\frac{\sqrt{7 (3 y^{2} - 5)}}{7})}^{2} + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $3 {(\frac{\sqrt{7 (3 y^{2} - 5)}}{7})}^{2} + 5 y^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Apply the product rule to $\frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ .

$3 (\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2

Simplify the numerator.

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Step 2.1.2.1.1.2.1

Rewrite ${\sqrt{7 (3 y^{2} - 5)}}^{2}$ as $7 (3 y^{2} - 5)$ .

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Step 2.1.2.1.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 (3 y^{2} - 5)}$ as ${(7 (3 y^{2} - 5))}^{\frac{1}{2}}$ .

$3 (\frac{{({(7 (3 y^{2} - 5))}^{\frac{1}{2}})}^{2}}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.1.2

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

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{1}{2} \cdot 2}}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.1.3

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

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{2}{2}}}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.2.1.4.1

Cancel the common factor.

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{2}{2}}}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.1.4.2

Rewrite the expression.

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.1.5

Simplify.

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$3 (\frac{7 (3 y^{2}) + 7 \cdot - 5}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.3

Multiply $3$ by $7$ .

$3 (\frac{21 y^{2} + 7 \cdot - 5}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.4

Multiply $7$ by $- 5$ .

$3 (\frac{21 y^{2} - 35}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.5

Factor $7$ out of $21 y^{2} - 35$ .

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Step 2.1.2.1.1.2.5.1

Factor $7$ out of $21 y^{2}$ .

$3 (\frac{7 (3 y^{2}) - 35}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.5.2

Factor $7$ out of $- 35$ .

$3 (\frac{7 (3 y^{2}) + 7 (- 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.2.5.3

Factor $7$ out of $7 (3 y^{2}) + 7 (- 5)$ .

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.3

Raise $7$ to the power of $2$ .

$3 (\frac{7 (3 y^{2} - 5)}{49}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.4

Cancel the common factors.

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Step 2.1.2.1.1.4.1

Factor $7$ out of $49$ .

$3 (\frac{7 (3 y^{2} - 5)}{7 \cdot 7}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.4.2

Cancel the common factor.

$3 (\frac{7 (3 y^{2} - 5)}{7 \cdot 7}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.4.3

Rewrite the expression.

$3 (\frac{3 y^{2} - 5}{7}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{3 y^{2} - 5}{7}) + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.1.5

Combine $3$ and $\frac{3 y^{2} - 5}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.2

To write $5 y^{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} \cdot \frac{7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.3

Simplify terms.

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Step 2.1.2.1.3.1

Combine $5 y^{2}$ and $\frac{7}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + \frac{5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{3 (3 y^{2} - 5) + 5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{3 (3 y^{2} - 5) + 5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

Apply the distributive property.

$\frac{3 (3 y^{2}) + 3 \cdot - 5 + 5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.4.2

Multiply $3$ by $3$ .

$\frac{9 y^{2} + 3 \cdot - 5 + 5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.4.3

Multiply $3$ by $- 5$ .

$\frac{9 y^{2} - 15 + 5 y^{2} \cdot 7}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.4.4

Multiply $7$ by $5$ .

$\frac{9 y^{2} - 15 + 35 y^{2}}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.1.2.1.4.5

Add $9 y^{2}$ and $35 y^{2}$ .

$\frac{44 y^{2} - 15}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2

Solve for $y$ in $\frac{44 y^{2} - 15}{7} = 12$ .

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Step 2.2.1

Multiply both sides by $7$ .

$\frac{44 y^{2} - 15}{7} \cdot 7 = 12 \cdot 7$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the left side.

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Step 2.2.2.1.1

Cancel the common factor of $7$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{44 y^{2} - 15}{7} \cdot 7 = 12 \cdot 7$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.2.1.1.2

Rewrite the expression.

$44 y^{2} - 15 = 12 \cdot 7$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 12 \cdot 7$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 12 \cdot 7$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

Multiply $12$ by $7$ .

$44 y^{2} - 15 = 84$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 84$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 84$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3

Solve for $y$ .

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Step 2.2.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.3.1.1

Add $15$ to both sides of the equation.

$44 y^{2} = 84 + 15$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.1.2

Add $84$ and $15$ .

$44 y^{2} = 99$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} = 99$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2

Divide each term in $44 y^{2} = 99$ by $44$ and simplify.

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Step 2.2.3.2.1

Divide each term in $44 y^{2} = 99$ by $44$ .

$\frac{44 y^{2}}{44} = \frac{99}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.2

Simplify the left side.

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Step 2.2.3.2.2.1

Cancel the common factor of $44$ .

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Step 2.2.3.2.2.1.1

Cancel the common factor.

$\frac{44 y^{2}}{44} = \frac{99}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.2.1.2

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

$y^{2} = \frac{99}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{99}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{99}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.3

Simplify the right side.

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Step 2.2.3.2.3.1

Cancel the common factor of $99$ and $44$ .

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Step 2.2.3.2.3.1.1

Factor $11$ out of $99$ .

$y^{2} = \frac{11 (9)}{44}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.3.1.2

Cancel the common factors.

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Step 2.2.3.2.3.1.2.1

Factor $11$ out of $44$ .

$y^{2} = \frac{11 \cdot 9}{11 \cdot 4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.3.1.2.2

Cancel the common factor.

$y^{2} = \frac{11 \cdot 9}{11 \cdot 4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.2.3.1.2.3

Rewrite the expression.

$y^{2} = \frac{9}{4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.3

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

$y = \pm \sqrt{\frac{9}{4}}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.4

Simplify $\pm \sqrt{\frac{9}{4}}$ .

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Step 2.2.3.4.1

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$y = \pm \frac{\sqrt{9}}{\sqrt{4}}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.4.2

Simplify the numerator.

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Step 2.2.3.4.2.1

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

$y = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.4.2.2

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

$y = \pm \frac{3}{\sqrt{4}}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{\sqrt{4}}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.4.3

Simplify the denominator.

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Step 2.2.3.4.3.1

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

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

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.4.3.2

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

$y = \pm \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.5

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

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Step 2.2.3.5.1

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

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

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.5.2

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

$y = - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.2.3.5.3

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

$y = \frac{3}{2}, - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 2.3

Replace all occurrences of $y$ with $\frac{3}{2}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ with $\frac{3}{2}$ .

$x = \frac{\sqrt{7 (3 {(\frac{3}{2})}^{2} - 5)}}{7}$

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\frac{\sqrt{7 (3 {(\frac{3}{2})}^{2} - 5)}}{7}$ .

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Step 2.3.2.1.1

Simplify the numerator.

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Step 2.3.2.1.1.1

Apply the product rule to $\frac{3}{2}$ .

$x = \frac{\sqrt{7 (3 (\frac{3^{2}}{2^{2}}) - 5)}}{7}$

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

Step 2.3.2.1.1.2

Raise $3$ to the power of $2$ .

$x = \frac{\sqrt{7 (3 (\frac{9}{2^{2}}) - 5)}}{7}$

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

Step 2.3.2.1.1.3

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

$x = \frac{\sqrt{7 (3 (\frac{9}{4}) - 5)}}{7}$

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

Step 2.3.2.1.1.4

Multiply $3 (\frac{9}{4})$ .

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Step 2.3.2.1.1.4.1

Combine $3$ and $\frac{9}{4}$ .

$x = \frac{\sqrt{7 (\frac{3 \cdot 9}{4} - 5)}}{7}$

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

Step 2.3.2.1.1.4.2

Multiply $3$ by $9$ .

$x = \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

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

$x = \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

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

Step 2.3.2.1.1.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{\sqrt{7 (\frac{27}{4} - 5 \cdot \frac{4}{4})}}{7}$

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

Step 2.3.2.1.1.6

Combine $- 5$ and $\frac{4}{4}$ .

$x = \frac{\sqrt{7 (\frac{27}{4} + \frac{- 5 \cdot 4}{4})}}{7}$

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

Step 2.3.2.1.1.7

Combine the numerators over the common denominator.

$x = \frac{\sqrt{7 (\frac{27 - 5 \cdot 4}{4})}}{7}$

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

Step 2.3.2.1.1.8

Simplify the numerator.

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Step 2.3.2.1.1.8.1

Multiply $- 5$ by $4$ .

$x = \frac{\sqrt{7 (\frac{27 - 20}{4})}}{7}$

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

Step 2.3.2.1.1.8.2

Subtract $20$ from $27$ .

$x = \frac{\sqrt{7 (\frac{7}{4})}}{7}$

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

$x = \frac{\sqrt{7 (\frac{7}{4})}}{7}$

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

Step 2.3.2.1.1.9

Combine $7$ and $\frac{7}{4}$ .

$x = \frac{\sqrt{\frac{7 \cdot 7}{4}}}{7}$

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

Step 2.3.2.1.1.10

Multiply $7$ by $7$ .

$x = \frac{\sqrt{\frac{49}{4}}}{7}$

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

Step 2.3.2.1.1.11

Rewrite $\sqrt{\frac{49}{4}}$ as $\frac{\sqrt{49}}{\sqrt{4}}$ .

$x = \frac{\frac{\sqrt{49}}{\sqrt{4}}}{7}$

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

Step 2.3.2.1.1.12

Simplify the numerator.

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Step 2.3.2.1.1.12.1

Rewrite $49$ as $7^{2}$ .

$x = \frac{\frac{\sqrt{7^{2}}}{\sqrt{4}}}{7}$

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

Step 2.3.2.1.1.12.2

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

$x = \frac{\frac{7}{\sqrt{4}}}{7}$

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

$x = \frac{\frac{7}{\sqrt{4}}}{7}$

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

Step 2.3.2.1.1.13

Simplify the denominator.

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Step 2.3.2.1.1.13.1

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

$x = \frac{\frac{7}{\sqrt{2^{2}}}}{7}$

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

Step 2.3.2.1.1.13.2

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

$x = \frac{\frac{7}{2}}{7}$

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

$x = \frac{\frac{7}{2}}{7}$

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

$x = \frac{\frac{7}{2}}{7}$

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

Step 2.3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7}{2} \cdot \frac{1}{7}$

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

Step 2.3.2.1.3

Cancel the common factor of $7$ .

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Step 2.3.2.1.3.1

Cancel the common factor.

$x = \frac{7}{2} \cdot \frac{1}{7}$

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

Step 2.3.2.1.3.2

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Step 2.4

Replace all occurrences of $y$ with $- \frac{3}{2}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ with $- \frac{3}{2}$ .

$x = \frac{\sqrt{7 (3 {(- \frac{3}{2})}^{2} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{\sqrt{7 (3 {(- \frac{3}{2})}^{2} - 5)}}{7}$ .

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Step 2.4.2.1.1

Simplify the numerator.

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Step 2.4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.2.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x = \frac{\sqrt{7 (3 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x = \frac{\sqrt{7 (3 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\sqrt{7 (3 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.2

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

$x = \frac{\sqrt{7 (3 (1 (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x = \frac{\sqrt{7 (3 (\frac{3^{2}}{2^{2}}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.4

Raise $3$ to the power of $2$ .

$x = \frac{\sqrt{7 (3 (\frac{9}{2^{2}}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.5

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

$x = \frac{\sqrt{7 (3 (\frac{9}{4}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.6

Multiply $3 (\frac{9}{4})$ .

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Step 2.4.2.1.1.6.1

Combine $3$ and $\frac{9}{4}$ .

$x = \frac{\sqrt{7 (\frac{3 \cdot 9}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.6.2

Multiply $3$ by $9$ .

$x = \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.7

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{\sqrt{7 (\frac{27}{4} - 5 \cdot \frac{4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.8

Combine $- 5$ and $\frac{4}{4}$ .

$x = \frac{\sqrt{7 (\frac{27}{4} + \frac{- 5 \cdot 4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.9

Combine the numerators over the common denominator.

$x = \frac{\sqrt{7 (\frac{27 - 5 \cdot 4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.10

Simplify the numerator.

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Step 2.4.2.1.1.10.1

Multiply $- 5$ by $4$ .

$x = \frac{\sqrt{7 (\frac{27 - 20}{4})}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.10.2

Subtract $20$ from $27$ .

$x = \frac{\sqrt{7 (\frac{7}{4})}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\sqrt{7 (\frac{7}{4})}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.11

Combine $7$ and $\frac{7}{4}$ .

$x = \frac{\sqrt{\frac{7 \cdot 7}{4}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.12

Multiply $7$ by $7$ .

$x = \frac{\sqrt{\frac{49}{4}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.13

Rewrite $\sqrt{\frac{49}{4}}$ as $\frac{\sqrt{49}}{\sqrt{4}}$ .

$x = \frac{\frac{\sqrt{49}}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.14

Simplify the numerator.

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Step 2.4.2.1.1.14.1

Rewrite $49$ as $7^{2}$ .

$x = \frac{\frac{\sqrt{7^{2}}}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.14.2

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

$x = \frac{\frac{7}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\frac{7}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.15

Simplify the denominator.

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Step 2.4.2.1.1.15.1

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

$x = \frac{\frac{7}{\sqrt{2^{2}}}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.1.15.2

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

$x = \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

$x = \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7}{2} \cdot \frac{1}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.3

Cancel the common factor of $7$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

$x = \frac{7}{2} \cdot \frac{1}{7}$

$y = - \frac{3}{2}$

Step 2.4.2.1.3.2

Rewrite the expression.

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

Step 3

Solve the system $x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}, 3 x^{2} + 5 y^{2} = 12$ .

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Step 3.1

Replace all occurrences of $x$ with $- \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $3 x^{2} + 5 y^{2} = 12$ with $- \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ .

$3 {(- \frac{\sqrt{7 (3 y^{2} - 5)}}{7})}^{2} + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $3 {(- \frac{\sqrt{7 (3 y^{2} - 5)}}{7})}^{2} + 5 y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.1.1.1.1

Apply the product rule to $- \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ .

$3 ({(- 1)}^{2} {(\frac{\sqrt{7 (3 y^{2} - 5)}}{7})}^{2}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.1.2

Apply the product rule to $\frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ .

$3 ({(- 1)}^{2} (\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}})) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 ({(- 1)}^{2} (\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}})) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.2

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

$3 (1 (\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}})) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.3

Multiply $\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}}$ by $1$ .

$3 (\frac{{\sqrt{7 (3 y^{2} - 5)}}^{2}}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4

Simplify the numerator.

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Step 3.1.2.1.1.4.1

Rewrite ${\sqrt{7 (3 y^{2} - 5)}}^{2}$ as $7 (3 y^{2} - 5)$ .

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Step 3.1.2.1.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 (3 y^{2} - 5)}$ as ${(7 (3 y^{2} - 5))}^{\frac{1}{2}}$ .

$3 (\frac{{({(7 (3 y^{2} - 5))}^{\frac{1}{2}})}^{2}}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.1.2

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

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{1}{2} \cdot 2}}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.1.3

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

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{2}{2}}}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.1.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.1.4.1

Cancel the common factor.

$3 (\frac{{(7 (3 y^{2} - 5))}^{\frac{2}{2}}}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.1.4.2

Rewrite the expression.

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.1.5

Simplify.

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.2

Apply the distributive property.

$3 (\frac{7 (3 y^{2}) + 7 \cdot - 5}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.3

Multiply $3$ by $7$ .

$3 (\frac{21 y^{2} + 7 \cdot - 5}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.4

Multiply $7$ by $- 5$ .

$3 (\frac{21 y^{2} - 35}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.5

Factor $7$ out of $21 y^{2} - 35$ .

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Step 3.1.2.1.1.4.5.1

Factor $7$ out of $21 y^{2}$ .

$3 (\frac{7 (3 y^{2}) - 35}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.5.2

Factor $7$ out of $- 35$ .

$3 (\frac{7 (3 y^{2}) + 7 (- 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.4.5.3

Factor $7$ out of $7 (3 y^{2}) + 7 (- 5)$ .

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{7 (3 y^{2} - 5)}{7^{2}}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.5

Raise $7$ to the power of $2$ .

$3 (\frac{7 (3 y^{2} - 5)}{49}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.6

Cancel the common factors.

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Step 3.1.2.1.1.6.1

Factor $7$ out of $49$ .

$3 (\frac{7 (3 y^{2} - 5)}{7 \cdot 7}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.6.2

Cancel the common factor.

$3 (\frac{7 (3 y^{2} - 5)}{7 \cdot 7}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.6.3

Rewrite the expression.

$3 (\frac{3 y^{2} - 5}{7}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$3 (\frac{3 y^{2} - 5}{7}) + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.1.7

Combine $3$ and $\frac{3 y^{2} - 5}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.2

To write $5 y^{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + 5 y^{2} \cdot \frac{7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.3

Simplify terms.

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Step 3.1.2.1.3.1

Combine $5 y^{2}$ and $\frac{7}{7}$ .

$\frac{3 (3 y^{2} - 5)}{7} + \frac{5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{3 (3 y^{2} - 5) + 5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{3 (3 y^{2} - 5) + 5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.4

Simplify the numerator.

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Step 3.1.2.1.4.1

Apply the distributive property.

$\frac{3 (3 y^{2}) + 3 \cdot - 5 + 5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.4.2

Multiply $3$ by $3$ .

$\frac{9 y^{2} + 3 \cdot - 5 + 5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.4.3

Multiply $3$ by $- 5$ .

$\frac{9 y^{2} - 15 + 5 y^{2} \cdot 7}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.4.4

Multiply $7$ by $5$ .

$\frac{9 y^{2} - 15 + 35 y^{2}}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.1.2.1.4.5

Add $9 y^{2}$ and $35 y^{2}$ .

$\frac{44 y^{2} - 15}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$\frac{44 y^{2} - 15}{7} = 12$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2

Solve for $y$ in $\frac{44 y^{2} - 15}{7} = 12$ .

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Step 3.2.1

Multiply both sides by $7$ .

$\frac{44 y^{2} - 15}{7} \cdot 7 = 12 \cdot 7$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{44 y^{2} - 15}{7} \cdot 7 = 12 \cdot 7$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.2.1.1.2

Rewrite the expression.

$44 y^{2} - 15 = 12 \cdot 7$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 12 \cdot 7$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 12 \cdot 7$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Multiply $12$ by $7$ .

$44 y^{2} - 15 = 84$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 84$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} - 15 = 84$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3

Solve for $y$ .

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Step 3.2.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.3.1.1

Add $15$ to both sides of the equation.

$44 y^{2} = 84 + 15$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.1.2

Add $84$ and $15$ .

$44 y^{2} = 99$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$44 y^{2} = 99$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2

Divide each term in $44 y^{2} = 99$ by $44$ and simplify.

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Step 3.2.3.2.1

Divide each term in $44 y^{2} = 99$ by $44$ .

$\frac{44 y^{2}}{44} = \frac{99}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.2

Simplify the left side.

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Step 3.2.3.2.2.1

Cancel the common factor of $44$ .

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Step 3.2.3.2.2.1.1

Cancel the common factor.

$\frac{44 y^{2}}{44} = \frac{99}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.2.1.2

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

$y^{2} = \frac{99}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{99}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{99}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.3

Simplify the right side.

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Step 3.2.3.2.3.1

Cancel the common factor of $99$ and $44$ .

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Step 3.2.3.2.3.1.1

Factor $11$ out of $99$ .

$y^{2} = \frac{11 (9)}{44}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.2.3.1.2.1

Factor $11$ out of $44$ .

$y^{2} = \frac{11 \cdot 9}{11 \cdot 4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.3.1.2.2

Cancel the common factor.

$y^{2} = \frac{11 \cdot 9}{11 \cdot 4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.2.3.1.2.3

Rewrite the expression.

$y^{2} = \frac{9}{4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y^{2} = \frac{9}{4}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.3

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

$y = \pm \sqrt{\frac{9}{4}}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.4

Simplify $\pm \sqrt{\frac{9}{4}}$ .

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Step 3.2.3.4.1

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$y = \pm \frac{\sqrt{9}}{\sqrt{4}}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.4.2

Simplify the numerator.

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Step 3.2.3.4.2.1

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

$y = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.4.2.2

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

$y = \pm \frac{3}{\sqrt{4}}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{\sqrt{4}}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.4.3

Simplify the denominator.

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Step 3.2.3.4.3.1

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

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

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.4.3.2

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

$y = \pm \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \pm \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.5

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

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Step 3.2.3.5.1

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

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

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.5.2

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

$y = - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.2.3.5.3

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

$y = \frac{3}{2}, - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

$y = \frac{3}{2}, - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$

Step 3.3

Replace all occurrences of $y$ with $\frac{3}{2}$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ with $\frac{3}{2}$ .

$x = - \frac{\sqrt{7 (3 {(\frac{3}{2})}^{2} - 5)}}{7}$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{\sqrt{7 (3 {(\frac{3}{2})}^{2} - 5)}}{7}$ .

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Step 3.3.2.1.1

Simplify the numerator.

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Step 3.3.2.1.1.1

Apply the product rule to $\frac{3}{2}$ .

$x = - \frac{\sqrt{7 (3 (\frac{3^{2}}{2^{2}}) - 5)}}{7}$

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

Step 3.3.2.1.1.2

Raise $3$ to the power of $2$ .

$x = - \frac{\sqrt{7 (3 (\frac{9}{2^{2}}) - 5)}}{7}$

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

Step 3.3.2.1.1.3

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

$x = - \frac{\sqrt{7 (3 (\frac{9}{4}) - 5)}}{7}$

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

Step 3.3.2.1.1.4

Multiply $3 (\frac{9}{4})$ .

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Step 3.3.2.1.1.4.1

Combine $3$ and $\frac{9}{4}$ .

$x = - \frac{\sqrt{7 (\frac{3 \cdot 9}{4} - 5)}}{7}$

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

Step 3.3.2.1.1.4.2

Multiply $3$ by $9$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

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

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

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

Step 3.3.2.1.1.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5 \cdot \frac{4}{4})}}{7}$

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

Step 3.3.2.1.1.6

Combine $- 5$ and $\frac{4}{4}$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} + \frac{- 5 \cdot 4}{4})}}{7}$

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

Step 3.3.2.1.1.7

Combine the numerators over the common denominator.

$x = - \frac{\sqrt{7 (\frac{27 - 5 \cdot 4}{4})}}{7}$

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

Step 3.3.2.1.1.8

Simplify the numerator.

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Step 3.3.2.1.1.8.1

Multiply $- 5$ by $4$ .

$x = - \frac{\sqrt{7 (\frac{27 - 20}{4})}}{7}$

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

Step 3.3.2.1.1.8.2

Subtract $20$ from $27$ .

$x = - \frac{\sqrt{7 (\frac{7}{4})}}{7}$

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

$x = - \frac{\sqrt{7 (\frac{7}{4})}}{7}$

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

Step 3.3.2.1.1.9

Combine $7$ and $\frac{7}{4}$ .

$x = - \frac{\sqrt{\frac{7 \cdot 7}{4}}}{7}$

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

Step 3.3.2.1.1.10

Multiply $7$ by $7$ .

$x = - \frac{\sqrt{\frac{49}{4}}}{7}$

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

Step 3.3.2.1.1.11

Rewrite $\sqrt{\frac{49}{4}}$ as $\frac{\sqrt{49}}{\sqrt{4}}$ .

$x = - \frac{\frac{\sqrt{49}}{\sqrt{4}}}{7}$

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

Step 3.3.2.1.1.12

Simplify the numerator.

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Step 3.3.2.1.1.12.1

Rewrite $49$ as $7^{2}$ .

$x = - \frac{\frac{\sqrt{7^{2}}}{\sqrt{4}}}{7}$

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

Step 3.3.2.1.1.12.2

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

$x = - \frac{\frac{7}{\sqrt{4}}}{7}$

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

$x = - \frac{\frac{7}{\sqrt{4}}}{7}$

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

Step 3.3.2.1.1.13

Simplify the denominator.

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Step 3.3.2.1.1.13.1

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

$x = - \frac{\frac{7}{\sqrt{2^{2}}}}{7}$

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

Step 3.3.2.1.1.13.2

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

$x = - \frac{\frac{7}{2}}{7}$

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

$x = - \frac{\frac{7}{2}}{7}$

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

$x = - \frac{\frac{7}{2}}{7}$

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

Step 3.3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = - (\frac{7}{2} \cdot \frac{1}{7})$

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

Step 3.3.2.1.3

Cancel the common factor of $7$ .

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Step 3.3.2.1.3.1

Cancel the common factor.

$x = - (\frac{7}{2} \cdot \frac{1}{7})$

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

Step 3.3.2.1.3.2

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Step 3.4

Replace all occurrences of $y$ with $- \frac{3}{2}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{7 (3 y^{2} - 5)}}{7}$ with $- \frac{3}{2}$ .

$x = - \frac{\sqrt{7 (3 {(- \frac{3}{2})}^{2} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \frac{\sqrt{7 (3 {(- \frac{3}{2})}^{2} - 5)}}{7}$ .

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.2.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x = - \frac{\sqrt{7 (3 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x = - \frac{\sqrt{7 (3 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\sqrt{7 (3 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.2

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

$x = - \frac{\sqrt{7 (3 (1 (\frac{3^{2}}{2^{2}})) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x = - \frac{\sqrt{7 (3 (\frac{3^{2}}{2^{2}}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.4

Raise $3$ to the power of $2$ .

$x = - \frac{\sqrt{7 (3 (\frac{9}{2^{2}}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.5

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

$x = - \frac{\sqrt{7 (3 (\frac{9}{4}) - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.6

Multiply $3 (\frac{9}{4})$ .

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Step 3.4.2.1.1.6.1

Combine $3$ and $\frac{9}{4}$ .

$x = - \frac{\sqrt{7 (\frac{3 \cdot 9}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.6.2

Multiply $3$ by $9$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5)}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.7

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} - 5 \cdot \frac{4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.8

Combine $- 5$ and $\frac{4}{4}$ .

$x = - \frac{\sqrt{7 (\frac{27}{4} + \frac{- 5 \cdot 4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.9

Combine the numerators over the common denominator.

$x = - \frac{\sqrt{7 (\frac{27 - 5 \cdot 4}{4})}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.10

Simplify the numerator.

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Step 3.4.2.1.1.10.1

Multiply $- 5$ by $4$ .

$x = - \frac{\sqrt{7 (\frac{27 - 20}{4})}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.10.2

Subtract $20$ from $27$ .

$x = - \frac{\sqrt{7 (\frac{7}{4})}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\sqrt{7 (\frac{7}{4})}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.11

Combine $7$ and $\frac{7}{4}$ .

$x = - \frac{\sqrt{\frac{7 \cdot 7}{4}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.12

Multiply $7$ by $7$ .

$x = - \frac{\sqrt{\frac{49}{4}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.13

Rewrite $\sqrt{\frac{49}{4}}$ as $\frac{\sqrt{49}}{\sqrt{4}}$ .

$x = - \frac{\frac{\sqrt{49}}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.14

Simplify the numerator.

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Step 3.4.2.1.1.14.1

Rewrite $49$ as $7^{2}$ .

$x = - \frac{\frac{\sqrt{7^{2}}}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.14.2

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

$x = - \frac{\frac{7}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\frac{7}{\sqrt{4}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.15

Simplify the denominator.

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Step 3.4.2.1.1.15.1

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

$x = - \frac{\frac{7}{\sqrt{2^{2}}}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.1.15.2

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

$x = - \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

$x = - \frac{\frac{7}{2}}{7}$

$y = - \frac{3}{2}$

Step 3.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = - (\frac{7}{2} \cdot \frac{1}{7})$

$y = - \frac{3}{2}$

Step 3.4.2.1.3

Cancel the common factor of $7$ .

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Step 3.4.2.1.3.1

Cancel the common factor.

$x = - (\frac{7}{2} \cdot \frac{1}{7})$

$y = - \frac{3}{2}$

Step 3.4.2.1.3.2

Rewrite the expression.

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

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

$y = - \frac{3}{2}$

Step 4

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

$(\frac{1}{2}, \frac{3}{2})$

$(\frac{1}{2}, - \frac{3}{2})$

$(- \frac{1}{2}, \frac{3}{2})$

$(- \frac{1}{2}, - \frac{3}{2})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, \frac{3}{2}), (\frac{1}{2}, - \frac{3}{2}), (- \frac{1}{2}, \frac{3}{2}), (- \frac{1}{2}, - \frac{3}{2})$

Equation Form:

$x = \frac{1}{2}, y = \frac{3}{2}$

$x = \frac{1}{2}, y = - \frac{3}{2}$

$x = - \frac{1}{2}, y = \frac{3}{2}$

$x = - \frac{1}{2}, y = - \frac{3}{2}$

Step 6