Finite Math Examples

$y = \frac{9}{x^{2} - 1}$

Step 1

Rewrite the equation as $\frac{9}{x^{2} - 1} = y$ .

$\frac{9}{x^{2} - 1} = y$

Step 2

Factor each term.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{9}{x^{2} - 1^{2}} = y$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{9}{(x + 1) (x - 1)} = y$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 1) (x - 1), 1$

Step 3.2

The LCM of one and any expression is the expression.

$(x + 1) (x - 1)$

Step 4

Multiply each term in $\frac{9}{(x + 1) (x - 1)} = y$ by $(x + 1) (x - 1)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{9}{(x + 1) (x - 1)} = y$ by $(x + 1) (x - 1)$ .

$\frac{9}{(x + 1) (x - 1)} ((x + 1) (x - 1)) = y ((x + 1) (x - 1))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $(x + 1) (x - 1)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{9}{(x + 1) (x - 1)} ((x + 1) (x - 1)) = y ((x + 1) (x - 1))$

Step 4.2.1.2

Rewrite the expression.

$9 = y ((x + 1) (x - 1))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$9 = y (x (x - 1) + 1 (x - 1))$

Step 4.3.1.2

Apply the distributive property.

$9 = y (x \cdot x + x \cdot - 1 + 1 (x - 1))$

Step 4.3.1.3

Apply the distributive property.

$9 = y (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1)$

Step 4.3.2

Simplify and combine like terms.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $x$ by $x$ .

$9 = y (x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1)$

Step 4.3.2.1.2

Move $- 1$ to the left of $x$ .

$9 = y (x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1)$

Step 4.3.2.1.3

Rewrite $- 1 x$ as $- x$ .

$9 = y (x^{2} - x + 1 x + 1 \cdot - 1)$

Step 4.3.2.1.4

Multiply $x$ by $1$ .

$9 = y (x^{2} - x + x + 1 \cdot - 1)$

Step 4.3.2.1.5

Multiply $- 1$ by $1$ .

$9 = y (x^{2} - x + x - 1)$

Step 4.3.2.2

Add $- x$ and $x$ .

$9 = y (x^{2} + 0 - 1)$

Step 4.3.2.3

Add $x^{2}$ and $0$ .

$9 = y (x^{2} - 1)$

Step 4.3.3

Simplify by multiplying through.

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Step 4.3.3.1

Apply the distributive property.

$9 = y x^{2} + y \cdot - 1$

Step 4.3.3.2

Move $- 1$ to the left of $y$ .

$9 = y x^{2} - 1 \cdot y$

Step 4.3.4

Rewrite $- 1 y$ as $- y$ .

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

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $y x^{2} - y = 9$ .

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

Step 5.2

Add $y$ to both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $y$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

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

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $y$ .

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Step 5.3.3.1.1

Cancel the common factor.

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

Step 5.3.3.1.2

Rewrite the expression.

$x^{2} = \frac{9}{y} + 1$

Step 5.4

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

$x = \pm \sqrt{\frac{9}{y} + 1}$

Step 5.5

Simplify $\pm \sqrt{\frac{9}{y} + 1}$ .

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Step 5.5.1

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

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

Step 5.5.2

Combine the numerators over the common denominator.

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

Step 5.5.3

Rewrite $\sqrt{\frac{9 + y}{y}}$ as $\frac{\sqrt{9 + y}}{\sqrt{y}}$ .

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

Step 5.5.4

Multiply $\frac{\sqrt{9 + y}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$x = \pm \frac{\sqrt{9 + y}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 5.5.5

Combine and simplify the denominator.

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Step 5.5.5.1

Multiply $\frac{\sqrt{9 + y}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 5.5.5.2

Raise $\sqrt{y}$ to the power of $1$ .

$x = \pm \frac{\sqrt{9 + y} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 5.5.5.3

Raise $\sqrt{y}$ to the power of $1$ .

$x = \pm \frac{\sqrt{9 + y} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 5.5.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{9 + y} \sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 5.5.5.5

Add $1$ and $1$ .

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

Step 5.5.5.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 5.5.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 5.5.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.5.5.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.5.5.6.4

Cancel the common factor of $2$ .

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Step 5.5.5.6.4.1

Cancel the common factor.

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

Step 5.5.5.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{9 + y} \sqrt{y}}{y^{1}}$

Step 5.5.5.6.5

Simplify.

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

Step 5.5.6

Combine using the product rule for radicals.

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

Step 5.5.7

Reorder factors in $\pm \frac{\sqrt{(9 + y) y}}{y}$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

$x = - \frac{\sqrt{y (9 + y)}}{y}$

Step 5.6.3

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

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

$x = - \frac{\sqrt{y (9 + y)}}{y}$

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

$x = - \frac{\sqrt{y (9 + y)}}{y}$

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

$x = - \frac{\sqrt{y (9 + y)}}{y}$