Algebra Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$y^{2} + 1, 1$

Step 2.2.2

Remove parentheses.

$y^{2} + 1, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$y^{2} + 1$

Step 2.3

Multiply each term in $\frac{2}{y^{2} + 1} = x$ by $y^{2} + 1$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{2}{y^{2} + 1} = x$ by $y^{2} + 1$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y^{2} + 1$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

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

Step 2.3.3.2

Multiply $x$ by $1$ .

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

Step 2.4

Solve the equation.

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Step 2.4.1

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

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

Step 2.4.2

Subtract $x$ from both sides of the equation.

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

Step 2.4.3

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

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Step 2.4.3.1

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

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

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $x$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.1.2

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

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

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.1

Cancel the common factor.

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

Step 2.4.3.3.1.2

Divide $- 1$ by $1$ .

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

Step 2.4.4

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

$y = \pm \sqrt{\frac{2}{x} - 1}$

Step 2.4.5

Simplify $\pm \sqrt{\frac{2}{x} - 1}$ .

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Step 2.4.5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y = \pm \sqrt{\frac{2}{x} - 1 \cdot \frac{x}{x}}$

Step 2.4.5.2

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 2.4.5.3

Combine the numerators over the common denominator.

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

Step 2.4.5.4

Rewrite $\sqrt{\frac{2 - x}{x}}$ as $\frac{\sqrt{2 - x}}{\sqrt{x}}$ .

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

Step 2.4.5.5

Multiply $\frac{\sqrt{2 - x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{2 - x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 2.4.5.6

Combine and simplify the denominator.

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Step 2.4.5.6.1

Multiply $\frac{\sqrt{2 - x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 2.4.5.6.2

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

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

Step 2.4.5.6.3

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

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

Step 2.4.5.6.4

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

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

Step 2.4.5.6.5

Add $1$ and $1$ .

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

Step 2.4.5.6.6

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

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Step 2.4.5.6.6.1

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

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

Step 2.4.5.6.6.2

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

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

Step 2.4.5.6.6.3

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

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

Step 2.4.5.6.6.4

Cancel the common factor of $2$ .

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Step 2.4.5.6.6.4.1

Cancel the common factor.

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

Step 2.4.5.6.6.4.2

Rewrite the expression.

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

Step 2.4.5.6.6.5

Simplify.

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

Step 2.4.5.7

Combine using the product rule for radicals.

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

Step 2.4.5.8

Reorder factors in $\pm \frac{\sqrt{(2 - x) x}}{x}$ .

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

Step 2.4.6

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

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Step 2.4.6.1

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