Finite Math Examples

$9 x^{2} + 25 y^{2} = 255$

Step 1

Subtract $25 y^{2}$ from both sides of the equation.

$9 x^{2} = 255 - 25 y^{2}$

Step 2

Divide each term in $9 x^{2} = 255 - 25 y^{2}$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} = 255 - 25 y^{2}$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{255}{9} + \frac{- 25 y^{2}}{9}$

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.

$\frac{9 x^{2}}{9} = \frac{255}{9} + \frac{- 25 y^{2}}{9}$

Step 2.2.1.2

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

$x^{2} = \frac{255}{9} + \frac{- 25 y^{2}}{9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $255$ and $9$ .

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Step 2.3.1.1.1

Factor $3$ out of $255$ .

$x^{2} = \frac{3 (85)}{9} + \frac{- 25 y^{2}}{9}$

Step 2.3.1.1.2

Cancel the common factors.

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Step 2.3.1.1.2.1

Factor $3$ out of $9$ .

$x^{2} = \frac{3 \cdot 85}{3 \cdot 3} + \frac{- 25 y^{2}}{9}$

Step 2.3.1.1.2.2

Cancel the common factor.

$x^{2} = \frac{3 \cdot 85}{3 \cdot 3} + \frac{- 25 y^{2}}{9}$

Step 2.3.1.1.2.3

Rewrite the expression.

$x^{2} = \frac{85}{3} + \frac{- 25 y^{2}}{9}$

Step 2.3.1.2

Move the negative in front of the fraction.

$x^{2} = \frac{85}{3} - \frac{25 y^{2}}{9}$

Step 3

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

$x = \pm \sqrt{\frac{85}{3} - \frac{25 y^{2}}{9}}$

Step 4

Simplify $\pm \sqrt{\frac{85}{3} - \frac{25 y^{2}}{9}}$ .

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Step 4.1

To write $\frac{85}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{85}{3} \cdot \frac{3}{3} - \frac{25 y^{2}}{9}}$

Step 4.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{85}{3}$ by $\frac{3}{3}$ .

$x = \pm \sqrt{\frac{85 \cdot 3}{3 \cdot 3} - \frac{25 y^{2}}{9}}$

Step 4.2.2

Multiply $3$ by $3$ .

$x = \pm \sqrt{\frac{85 \cdot 3}{9} - \frac{25 y^{2}}{9}}$

Step 4.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{85 \cdot 3 - 25 y^{2}}{9}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor $5$ out of $85 \cdot 3 - 25 y^{2}$ .

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Step 4.4.1.1

Factor $5$ out of $85 \cdot 3$ .

$x = \pm \sqrt{\frac{5 (17 \cdot 3) - 25 y^{2}}{9}}$

Step 4.4.1.2

Factor $5$ out of $- 25 y^{2}$ .

$x = \pm \sqrt{\frac{5 (17 \cdot 3) + 5 (- 5 y^{2})}{9}}$

Step 4.4.1.3

Factor $5$ out of $5 (17 \cdot 3) + 5 (- 5 y^{2})$ .

$x = \pm \sqrt{\frac{5 (17 \cdot 3 - 5 y^{2})}{9}}$

Step 4.4.2

Multiply $17$ by $3$ .

$x = \pm \sqrt{\frac{5 (51 - 5 y^{2})}{9}}$

Step 4.5

Rewrite $\frac{5 (51 - 5 y^{2})}{9}$ as ${(\frac{1}{3})}^{2} (5 (51 - 5 y^{2}))$ .

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Step 4.5.1

Factor the perfect power $1^{2}$ out of $5 (51 - 5 y^{2})$ .

$x = \pm \sqrt{\frac{1^{2} (5 (51 - 5 y^{2}))}{9}}$

Step 4.5.2

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

$x = \pm \sqrt{\frac{1^{2} (5 (51 - 5 y^{2}))}{3^{2} \cdot 1}}$

Step 4.5.3

Rearrange the fraction $\frac{1^{2} (5 (51 - 5 y^{2}))}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{3})}^{2} (5 (51 - 5 y^{2}))}$

Step 4.6

Pull terms out from under the radical.

$x = \pm \frac{1}{3} \sqrt{5 (51 - 5 y^{2})}$

Step 4.7

Combine $\frac{1}{3}$ and $\sqrt{5 (51 - 5 y^{2})}$ .

$x = \pm \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

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.

$x = \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

Step 5.2

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

$x = - \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

Step 5.3

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

$x = \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

$x = - \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

$x = \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$

$x = - \frac{\sqrt{5 (51 - 5 y^{2})}}{3}$