Linear Algebra Examples

$z = \frac{y^{2}}{16} - \frac{x^{2}}{25}$

Step 1

Rewrite the equation as $\frac{y^{2}}{16} - \frac{x^{2}}{25} = z$ .

$\frac{y^{2}}{16} - \frac{x^{2}}{25} = z$

Step 2

Add $\frac{x^{2}}{25}$ to both sides of the equation.

$\frac{y^{2}}{16} = z + \frac{x^{2}}{25}$

Step 3

Multiply both sides of the equation by $16$ .

$16 \frac{y^{2}}{16} = 16 (z + \frac{x^{2}}{25})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $16$ .

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Step 4.1.1.1

Cancel the common factor.

$16 \frac{y^{2}}{16} = 16 (z + \frac{x^{2}}{25})$

Step 4.1.1.2

Rewrite the expression.

$y^{2} = 16 (z + \frac{x^{2}}{25})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $16 (z + \frac{x^{2}}{25})$ .

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Step 4.2.1.1

Apply the distributive property.

$y^{2} = 16 z + 16 \frac{x^{2}}{25}$

Step 4.2.1.2

Combine $16$ and $\frac{x^{2}}{25}$ .

$y^{2} = 16 z + \frac{16 x^{2}}{25}$

Step 5

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

$y = \pm \sqrt{16 z + \frac{16 x^{2}}{25}}$

Step 6

Simplify $\pm \sqrt{16 z + \frac{16 x^{2}}{25}}$ .

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Step 6.1

Factor $16$ out of $16 z + \frac{16 x^{2}}{25}$ .

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Step 6.1.1

Factor $16$ out of $16 z$ .

$y = \pm \sqrt{16 (z) + \frac{16 x^{2}}{25}}$

Step 6.1.2

Factor $16$ out of $\frac{16 x^{2}}{25}$ .

$y = \pm \sqrt{16 (z) + 16 \frac{x^{2}}{25}}$

Step 6.1.3

Factor $16$ out of $16 (z) + 16 \frac{x^{2}}{25}$ .

$y = \pm \sqrt{16 (z + \frac{x^{2}}{25})}$

Step 6.2

To write $z$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = \pm \sqrt{16 (z \cdot \frac{25}{25} + \frac{x^{2}}{25})}$

Step 6.3

Simplify terms.

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Step 6.3.1

Combine $z$ and $\frac{25}{25}$ .

$y = \pm \sqrt{16 (\frac{z \cdot 25}{25} + \frac{x^{2}}{25})}$

Step 6.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{16 \frac{z \cdot 25 + x^{2}}{25}}$

Step 6.4

Move $25$ to the left of $z$ .

$y = \pm \sqrt{16 \frac{25 z + x^{2}}{25}}$

Step 6.5

Combine $16$ and $\frac{25 z + x^{2}}{25}$ .

$y = \pm \sqrt{\frac{16 (25 z + x^{2})}{25}}$

Step 6.6

Rewrite $\frac{16 (25 z + x^{2})}{25}$ as ${(\frac{2^{2}}{5})}^{2} (25 z + x^{2})$ .

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Step 6.6.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (25 z + x^{2})$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} (25 z + x^{2})}{25}}$

Step 6.6.2

Factor the perfect power $5^{2}$ out of $25$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} (25 z + x^{2})}{5^{2} \cdot 1}}$

Step 6.6.3

Rearrange the fraction $\frac{{(2^{2})}^{2} (25 z + x^{2})}{5^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{2^{2}}{5})}^{2} (25 z + x^{2})}$

Step 6.7

Pull terms out from under the radical.

$y = \pm \frac{2^{2}}{5} \sqrt{25 z + x^{2}}$

Step 6.8

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

$y = \pm \frac{4}{5} \sqrt{25 z + x^{2}}$

Step 6.9

Combine $\frac{4}{5}$ and $\sqrt{25 z + x^{2}}$ .

$y = \pm \frac{4 \sqrt{25 z + x^{2}}}{5}$

Step 7

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

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Step 7.1

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

$y = \frac{4 \sqrt{25 z + x^{2}}}{5}$

Step 7.2

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

$y = - \frac{4 \sqrt{25 z + x^{2}}}{5}$

Step 7.3

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

$y = \frac{4 \sqrt{25 z + x^{2}}}{5}$

$y = - \frac{4 \sqrt{25 z + x^{2}}}{5}$

$y = \frac{4 \sqrt{25 z + x^{2}}}{5}$

$y = - \frac{4 \sqrt{25 z + x^{2}}}{5}$

Step 8

Set the radicand in $\sqrt{25 z + x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$25 z + x^{2} \geq 0$

Step 9

Solve for $z$ .

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Step 9.1

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

$25 z \geq - x^{2}$

Step 9.2

Divide each term in $25 z \geq - x^{2}$ by $25$ and simplify.

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Step 9.2.1

Divide each term in $25 z \geq - x^{2}$ by $25$ .

$\frac{25 z}{25} \geq \frac{- x^{2}}{25}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $25$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{25 z}{25} \geq \frac{- x^{2}}{25}$

Step 9.2.2.1.2

Divide $z$ by $1$ .

$z \geq \frac{- x^{2}}{25}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Move the negative in front of the fraction.

$z \geq - \frac{x^{2}}{25}$

Step 10

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${z | z \in ℝ}$