Linear Algebra Examples

$- 16 x^{2} + 9 y^{2} - 144 = 0$

Step 1

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

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Step 1.1

Add $16 x^{2}$ to both sides of the equation.

$9 y^{2} - 144 = 16 x^{2}$

Step 1.2

Add $144$ to both sides of the equation.

$9 y^{2} = 16 x^{2} + 144$

Step 2

Divide each term in $9 y^{2} = 16 x^{2} + 144$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 y^{2} = 16 x^{2} + 144$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{16 x^{2}}{9} + \frac{144}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{16 x^{2}}{9} + \frac{144}{9}$

Step 2.2.1.2

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

$y^{2} = \frac{16 x^{2}}{9} + \frac{144}{9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $144$ by $9$ .

$y^{2} = \frac{16 x^{2}}{9} + 16$

Step 3

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

$y = \pm \sqrt{\frac{16 x^{2}}{9} + 16}$

Step 4

Simplify $\pm \sqrt{\frac{16 x^{2}}{9} + 16}$ .

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Step 4.1

Factor $16$ out of $\frac{16 x^{2}}{9} + 16$ .

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Step 4.1.1

Factor $16$ out of $\frac{16 x^{2}}{9}$ .

$y = \pm \sqrt{16 \frac{x^{2}}{9} + 16}$

Step 4.1.2

Factor $16$ out of $16$ .

$y = \pm \sqrt{16 \frac{x^{2}}{9} + 16 (1)}$

Step 4.1.3

Factor $16$ out of $16 \frac{x^{2}}{9} + 16 (1)$ .

$y = \pm \sqrt{16 (\frac{x^{2}}{9} + 1)}$

Step 4.2

Write $1$ as a fraction with a common denominator.

$y = \pm \sqrt{16 (\frac{x^{2}}{9} + \frac{9}{9})}$

Step 4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{16 \frac{x^{2} + 9}{9}}$

Step 4.4

Combine $16$ and $\frac{x^{2} + 9}{9}$ .

$y = \pm \sqrt{\frac{16 (x^{2} + 9)}{9}}$

Step 4.5

Rewrite $\frac{16 (x^{2} + 9)}{9}$ as ${(\frac{2^{2}}{3})}^{2} (x^{2} + 9)$ .

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Step 4.5.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (x^{2} + 9)$ .

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

Step 4.5.2

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

$y = \pm \sqrt{\frac{{(2^{2})}^{2} (x^{2} + 9)}{3^{2} \cdot 1}}$

Step 4.5.3

Rearrange the fraction $\frac{{(2^{2})}^{2} (x^{2} + 9)}{3^{2} \cdot 1}$ .

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

Step 4.6

Pull terms out from under the radical.

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

Step 4.7

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

$y = \pm \frac{4}{3} \sqrt{x^{2} + 9}$

Step 4.8

Combine $\frac{4}{3}$ and $\sqrt{x^{2} + 9}$ .

$y = \pm \frac{4 \sqrt{x^{2} + 9}}{3}$

Step 5

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

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Step 5.1

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

$y = \frac{4 \sqrt{x^{2} + 9}}{3}$

Step 5.2

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

$y = - \frac{4 \sqrt{x^{2} + 9}}{3}$

Step 5.3

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

$y = \frac{4 \sqrt{x^{2} + 9}}{3}$

$y = - \frac{4 \sqrt{x^{2} + 9}}{3}$

$y = \frac{4 \sqrt{x^{2} + 9}}{3}$

$y = - \frac{4 \sqrt{x^{2} + 9}}{3}$

Step 6

Set the radicand in $\sqrt{x^{2} + 9}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 9 \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $9$ from both sides of the inequality.

$x^{2} \geq - 9$

Step 7.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 8

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 9