Trigonometry Examples

$4 x^{2} + 9 y^{2} = 36$

Step 1

Solve the equation for $y$ .

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Step 1.1

Subtract $4 x^{2}$ from both sides of the equation.

$9 y^{2} = 36 - 4 x^{2}$

Step 1.2

Divide each term in $9 y^{2} = 36 - 4 x^{2}$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 y^{2} = 36 - 4 x^{2}$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{36}{9} + \frac{- 4 x^{2}}{9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{36}{9} + \frac{- 4 x^{2}}{9}$

Step 1.2.2.1.2

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

$y^{2} = \frac{36}{9} + \frac{- 4 x^{2}}{9}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $36$ by $9$ .

$y^{2} = 4 + \frac{- 4 x^{2}}{9}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

Step 1.3

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

$y = \pm \sqrt{4 - \frac{4 x^{2}}{9}}$

Step 1.4

Simplify $\pm \sqrt{4 - \frac{4 x^{2}}{9}}$ .

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Step 1.4.1

Write the expression using exponents.

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Step 1.4.1.1

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

$y = \pm \sqrt{2^{2} - \frac{4 x^{2}}{9}}$

Step 1.4.1.2

Rewrite $\frac{4 x^{2}}{9}$ as ${(\frac{2 x}{3})}^{2}$ .

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

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = \frac{2 x}{3}$ .

$y = \pm \sqrt{(2 + \frac{2 x}{3}) (2 - \frac{2 x}{3})}$

Step 1.4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.4.4

Combine $2$ and $\frac{3}{3}$ .

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

Step 1.4.5

Combine the numerators over the common denominator.

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

Step 1.4.6

Factor $2$ out of $2 \cdot 3 + 2 x$ .

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Step 1.4.6.1

Factor $2$ out of $2 \cdot 3$ .

$y = \pm \sqrt{\frac{2 (3) + 2 x}{3} (2 - \frac{2 x}{3})}$

Step 1.4.6.2

Factor $2$ out of $2 x$ .

$y = \pm \sqrt{\frac{2 (3) + 2 (x)}{3} (2 - \frac{2 x}{3})}$

Step 1.4.6.3

Factor $2$ out of $2 (3) + 2 (x)$ .

$y = \pm \sqrt{\frac{2 (3 + x)}{3} (2 - \frac{2 x}{3})}$

Step 1.4.7

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.4.8

Combine $2$ and $\frac{3}{3}$ .

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

Step 1.4.9

Combine the numerators over the common denominator.

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

Step 1.4.10

Factor $2$ out of $2 \cdot 3 - 2 x$ .

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Step 1.4.10.1

Factor $2$ out of $2 \cdot 3$ .

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

Step 1.4.10.2

Factor $2$ out of $- 2 x$ .

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

Step 1.4.10.3

Factor $2$ out of $2 (3) + 2 (- x)$ .

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

Step 1.4.11

Multiply $\frac{2 (3 + x)}{3}$ by $\frac{2 (3 - x)}{3}$ .

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

Step 1.4.12

Multiply.

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Step 1.4.12.1

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{4 (3 + x) (3 - x)}{3 \cdot 3}}$

Step 1.4.12.2

Multiply $3$ by $3$ .

$y = \pm \sqrt{\frac{4 (3 + x) (3 - x)}{9}}$

Step 1.4.13

Rewrite $\frac{4 (3 + x) (3 - x)}{9}$ as ${(\frac{2}{3})}^{2} ((3 + x) (3 - x))$ .

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Step 1.4.13.1

Factor the perfect power $2^{2}$ out of $4 (3 + x) (3 - x)$ .

$y = \pm \sqrt{\frac{2^{2} ((3 + x) (3 - x))}{9}}$

Step 1.4.13.2

Factor the perfect power $3^{2}$ out of $9$ .

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

Step 1.4.13.3

Rearrange the fraction $\frac{2^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

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

Step 1.4.14

Pull terms out from under the radical.

$y = \pm \frac{2}{3} \sqrt{(3 + x) (3 - x)}$

Step 1.4.15

Combine $\frac{2}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$y = \pm \frac{2 \sqrt{(3 + x) (3 - x)}}{3}$

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.

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

Step 1.5.2

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

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

Step 1.5.3

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

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

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

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

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

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

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

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degree of the variable in the equation violates the linear equation definition, which means that the equation is not a linear equation.

Not Linear