Finite Math Examples

$\frac{{(X - 6)}^{2}}{100} + \frac{{(y - 7)}^{2}}{121} = 1$

Step 1

Subtract $\frac{{(y - 7)}^{2}}{121}$ from both sides of the equation.

$\frac{{(X - 6)}^{2}}{100} = 1 - \frac{{(y - 7)}^{2}}{121}$

Step 2

Simplify $1 - \frac{{(y - 7)}^{2}}{121}$ .

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Step 2.1

Combine into one fraction.

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Step 2.1.1

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

$\frac{{(X - 6)}^{2}}{100} = \frac{121}{121} - \frac{{(y - 7)}^{2}}{121}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(X - 6)}^{2}}{100} = \frac{121 - {(y - 7)}^{2}}{121}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $121$ as $11^{2}$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{11^{2} - {(y - 7)}^{2}}{121}$

Step 2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 11$ and $b = y - 7$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(11 + y - 7) (11 - (y - 7))}{121}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Subtract $7$ from $11$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (11 - (y - 7))}{121}$

Step 2.2.3.2

Apply the distributive property.

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (11 - y - - 7)}{121}$

Step 2.2.3.3

Multiply $- 1$ by $- 7$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (11 - y + 7)}{121}$

Step 2.2.3.4

Add $11$ and $7$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (- y + 18)}{121}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- y$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (- (y) + 18)}{121}$

Step 2.3.2

Rewrite $18$ as $- 1 (- 18)$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (- (y) - 1 (- 18))}{121}$

Step 2.3.3

Factor $- 1$ out of $- (y) - 1 (- 18)$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (- (y - 18))}{121}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Rewrite $- (y - 18)$ as $- 1 (y - 18)$ .

$\frac{{(X - 6)}^{2}}{100} = \frac{(y + 4) (- 1 (y - 18))}{121}$

Step 2.3.4.2

Move the negative in front of the fraction.

$\frac{{(X - 6)}^{2}}{100} = - \frac{(y + 4) (y - 18)}{121}$

Step 3

Multiply both sides of the equation by $100$ .

$100 \frac{{(X - 6)}^{2}}{100} = 100 (- \frac{(y + 4) (y - 18)}{121})$

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 $100$ .

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Step 4.1.1.1

Cancel the common factor.

$100 \frac{{(X - 6)}^{2}}{100} = 100 (- \frac{(y + 4) (y - 18)}{121})$

Step 4.1.1.2

Rewrite the expression.

${(X - 6)}^{2} = 100 (- \frac{(y + 4) (y - 18)}{121})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $100 (- \frac{(y + 4) (y - 18)}{121})$ .

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Step 4.2.1.1

Multiply $100 (- \frac{(y + 4) (y - 18)}{121})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $100$ .

${(X - 6)}^{2} = - 100 \frac{(y + 4) (y - 18)}{121}$

Step 4.2.1.1.2

Combine $- 100$ and $\frac{(y + 4) (y - 18)}{121}$ .

${(X - 6)}^{2} = \frac{- 100 ((y + 4) (y - 18))}{121}$

Step 4.2.1.2

Move the negative in front of the fraction.

${(X - 6)}^{2} = - \frac{100 (y + 4) (y - 18)}{121}$

Step 5

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

$X - 6 = \pm \sqrt{- \frac{100 (y + 4) (y - 18)}{121}}$

Step 6

Simplify $\pm \sqrt{- \frac{100 (y + 4) (y - 18)}{121}}$ .

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Step 6.1

Rewrite $- \frac{100 (y + 4) (y - 18)}{121}$ as ${(\frac{10}{11})}^{2} \cdot (- (y + 2^{2}) (y - 18))$ .

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Step 6.1.1

Factor the perfect power $10^{2}$ out of $100 (y + 4) (y - 18)$ .

$X - 6 = \pm \sqrt{- \frac{10^{2} ((y + 2^{2}) (y - 18))}{121}}$

Step 6.1.2

Factor the perfect power $11^{2}$ out of $121$ .

$X - 6 = \pm \sqrt{- \frac{10^{2} ((y + 2^{2}) (y - 18))}{11^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{10^{2} ((y + 2^{2}) (y - 18))}{11^{2} \cdot 1}$ .

$X - 6 = \pm \sqrt{- ({(\frac{10}{11})}^{2} ((y + 2^{2}) (y - 18)))}$

Step 6.1.4

Reorder $- 1$ and ${(\frac{10}{11})}^{2}$ .

$X - 6 = \pm \sqrt{{(\frac{10}{11})}^{2} \cdot - 1 (y + 2^{2}) (y - 18)}$

Step 6.1.5

Add parentheses.

$X - 6 = \pm \sqrt{{(\frac{10}{11})}^{2} \cdot - 1 ((y + 2^{2}) (y - 18))}$

Step 6.1.6

Add parentheses.

$X - 6 = \pm \sqrt{{(\frac{10}{11})}^{2} \cdot (- (y + 2^{2}) (y - 18))}$

Step 6.2

Pull terms out from under the radical.

$X - 6 = \pm \frac{10}{11} \sqrt{- (y + 2^{2}) (y - 18)}$

Step 6.3

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

$X - 6 = \pm \frac{10}{11} \sqrt{- (y + 4) (y - 18)}$

Step 6.4

Combine $\frac{10}{11}$ and $\sqrt{- (y + 4) (y - 18)}$ .

$X - 6 = \pm \frac{10 \sqrt{- (y + 4) (y - 18)}}{11}$

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.

$X - 6 = \frac{10 \sqrt{- (y + 4) (y - 18)}}{11}$

Step 7.2

Add $6$ to both sides of the equation.

$X = \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$

Step 7.3

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

$X - 6 = - \frac{10 \sqrt{- (y + 4) (y - 18)}}{11}$

Step 7.4

Add $6$ to both sides of the equation.

$X = - \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$

Step 7.5

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

$X = \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$

$X = - \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$

$X = \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$

$X = - \frac{10 \sqrt{- (y + 4) (y - 18)}}{11} + 6$