Finite Math Examples

$\frac{x^{2}}{100} + \frac{y^{2}}{49} = 1$

Step 1

Subtract $\frac{y^{2}}{49}$ from both sides of the equation.

$\frac{x^{2}}{100} = 1 - \frac{y^{2}}{49}$

Step 2

Multiply both sides of the equation by $100$ .

$100 \frac{x^{2}}{100} = 100 (1 - \frac{y^{2}}{49})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $100$ .

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Step 3.1.1.1

Cancel the common factor.

$100 \frac{x^{2}}{100} = 100 (1 - \frac{y^{2}}{49})$

Step 3.1.1.2

Rewrite the expression.

$x^{2} = 100 (1 - \frac{y^{2}}{49})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $100 (1 - \frac{y^{2}}{49})$ .

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Step 3.2.1.1

Apply the distributive property.

$x^{2} = 100 \cdot 1 + 100 (- \frac{y^{2}}{49})$

Step 3.2.1.2

Multiply $100$ by $1$ .

$x^{2} = 100 + 100 (- \frac{y^{2}}{49})$

Step 3.2.1.3

Multiply $100 (- \frac{y^{2}}{49})$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $100$ .

$x^{2} = 100 - 100 \frac{y^{2}}{49}$

Step 3.2.1.3.2

Combine $- 100$ and $\frac{y^{2}}{49}$ .

$x^{2} = 100 + \frac{- 100 y^{2}}{49}$

Step 3.2.1.4

Move the negative in front of the fraction.

$x^{2} = 100 - \frac{100 y^{2}}{49}$

Step 4

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

$x = \pm \sqrt{100 - \frac{100 y^{2}}{49}}$

Step 5

Simplify $\pm \sqrt{100 - \frac{100 y^{2}}{49}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2} - \frac{100 y^{2}}{49}}$

Step 5.1.2

Rewrite $\frac{100 y^{2}}{49}$ as ${(\frac{10 y}{7})}^{2}$ .

$x = \pm \sqrt{10^{2} - {(\frac{10 y}{7})}^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = \frac{10 y}{7}$ .

$x = \pm \sqrt{(10 + \frac{10 y}{7}) (10 - \frac{10 y}{7})}$

Step 5.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \pm \sqrt{(10 \cdot \frac{7}{7} + \frac{10 y}{7}) (10 - \frac{10 y}{7})}$

Step 5.4

Combine $10$ and $\frac{7}{7}$ .

$x = \pm \sqrt{(\frac{10 \cdot 7}{7} + \frac{10 y}{7}) (10 - \frac{10 y}{7})}$

Step 5.5

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{10 \cdot 7 + 10 y}{7} (10 - \frac{10 y}{7})}$

Step 5.6

Factor $10$ out of $10 \cdot 7 + 10 y$ .

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Step 5.6.1

Factor $10$ out of $10 \cdot 7$ .

$x = \pm \sqrt{\frac{10 (7) + 10 y}{7} (10 - \frac{10 y}{7})}$

Step 5.6.2

Factor $10$ out of $10 y$ .

$x = \pm \sqrt{\frac{10 (7) + 10 (y)}{7} (10 - \frac{10 y}{7})}$

Step 5.6.3

Factor $10$ out of $10 (7) + 10 (y)$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} (10 - \frac{10 y}{7})}$

Step 5.7

To write $10$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} (10 \cdot \frac{7}{7} - \frac{10 y}{7})}$

Step 5.8

Combine $10$ and $\frac{7}{7}$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} (\frac{10 \cdot 7}{7} - \frac{10 y}{7})}$

Step 5.9

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{10 (7 + y)}{7} \cdot \frac{10 \cdot 7 - 10 y}{7}}$

Step 5.10

Factor $10$ out of $10 \cdot 7 - 10 y$ .

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Step 5.10.1

Factor $10$ out of $10 \cdot 7$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} \cdot \frac{10 (7) - 10 y}{7}}$

Step 5.10.2

Factor $10$ out of $- 10 y$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} \cdot \frac{10 (7) + 10 (- y)}{7}}$

Step 5.10.3

Factor $10$ out of $10 (7) + 10 (- y)$ .

$x = \pm \sqrt{\frac{10 (7 + y)}{7} \cdot \frac{10 (7 - y)}{7}}$

Step 5.11

Multiply $\frac{10 (7 + y)}{7}$ by $\frac{10 (7 - y)}{7}$ .

$x = \pm \sqrt{\frac{10 (7 + y) (10 (7 - y))}{7 \cdot 7}}$

Step 5.12

Multiply.

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Step 5.12.1

Multiply $10$ by $10$ .

$x = \pm \sqrt{\frac{100 (7 + y) (7 - y)}{7 \cdot 7}}$

Step 5.12.2

Multiply $7$ by $7$ .

$x = \pm \sqrt{\frac{100 (7 + y) (7 - y)}{49}}$

Step 5.13

Rewrite $\frac{100 (7 + y) (7 - y)}{49}$ as ${(\frac{10}{7})}^{2} ((7 + y) (7 - y))$ .

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Step 5.13.1

Factor the perfect power $10^{2}$ out of $100 (7 + y) (7 - y)$ .

$x = \pm \sqrt{\frac{10^{2} ((7 + y) (7 - y))}{49}}$

Step 5.13.2

Factor the perfect power $7^{2}$ out of $49$ .

$x = \pm \sqrt{\frac{10^{2} ((7 + y) (7 - y))}{7^{2} \cdot 1}}$

Step 5.13.3

Rearrange the fraction $\frac{10^{2} ((7 + y) (7 - y))}{7^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{10}{7})}^{2} ((7 + y) (7 - y))}$

Step 5.14

Pull terms out from under the radical.

$x = \pm \frac{10}{7} \sqrt{(7 + y) (7 - y)}$

Step 5.15

Combine $\frac{10}{7}$ and $\sqrt{(7 + y) (7 - y)}$ .

$x = \pm \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

Step 6

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

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Step 6.1

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

$x = \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

Step 6.2

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

$x = - \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

Step 6.3

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

$x = \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

$x = - \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

$x = \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$

$x = - \frac{10 \sqrt{(7 + y) (7 - y)}}{7}$