Precalculus Examples

$f (x) = \frac{x^{2} - 6}{7 x^{2}}$

Step 1

Write $f (x) = \frac{x^{2} - 6}{7 x^{2}}$ as an equation.

$y = \frac{x^{2} - 6}{7 x^{2}}$

Step 2

Interchange the variables.

$x = \frac{y^{2} - 6}{7 y^{2}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{2} - 6}{7 y^{2}} = x$ .

$\frac{y^{2} - 6}{7 y^{2}} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$7 y^{2}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$7 y^{2}$

Step 3.3

Multiply each term in $\frac{y^{2} - 6}{7 y^{2}} = x$ by $7 y^{2}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{y^{2} - 6}{7 y^{2}} = x$ by $7 y^{2}$ .

$\frac{y^{2} - 6}{7 y^{2}} (7 y^{2}) = x (7 y^{2})$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$7 \frac{y^{2} - 6}{7 y^{2}} y^{2} = x (7 y^{2})$

Step 3.3.2.2

Cancel the common factor of $7$ .

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Step 3.3.2.2.1

Factor $7$ out of $7 y^{2}$ .

$7 \frac{y^{2} - 6}{7 (y^{2})} y^{2} = x (7 y^{2})$

Step 3.3.2.2.2

Cancel the common factor.

$7 \frac{y^{2} - 6}{7 y^{2}} y^{2} = x (7 y^{2})$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{y^{2} - 6}{y^{2}} y^{2} = x (7 y^{2})$

Step 3.3.2.3

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

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{y^{2} - 6}{y^{2}} y^{2} = x (7 y^{2})$

Step 3.3.2.3.2

Rewrite the expression.

$y^{2} - 6 = x (7 y^{2})$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Rewrite using the commutative property of multiplication.

$y^{2} - 6 = 7 x y^{2}$

Step 3.4

Solve the equation.

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Step 3.4.1

Subtract $7 x y^{2}$ from both sides of the equation.

$y^{2} - 6 - 7 x y^{2} = 0$

Step 3.4.2

Add $6$ to both sides of the equation.

$y^{2} - 7 x y^{2} = 6$

Step 3.4.3

Factor $y^{2}$ out of $y^{2} - 7 x y^{2}$ .

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Step 3.4.3.1

Multiply by $1$ .

$y^{2} \cdot 1 - 7 x y^{2} = 6$

Step 3.4.3.2

Factor $y^{2}$ out of $- 7 x y^{2}$ .

$y^{2} \cdot 1 + y^{2} (- 7 x) = 6$

Step 3.4.3.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} (- 7 x)$ .

$y^{2} (1 - 7 x) = 6$

Step 3.4.4

Divide each term in $y^{2} (1 - 7 x) = 6$ by $1 - 7 x$ and simplify.

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Step 3.4.4.1

Divide each term in $y^{2} (1 - 7 x) = 6$ by $1 - 7 x$ .

$\frac{y^{2} (1 - 7 x)}{1 - 7 x} = \frac{6}{1 - 7 x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $1 - 7 x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y^{2} (1 - 7 x)}{1 - 7 x} = \frac{6}{1 - 7 x}$

Step 3.4.4.2.1.2

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

$y^{2} = \frac{6}{1 - 7 x}$

Step 3.4.5

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

$y = \pm \sqrt{\frac{6}{1 - 7 x}}$

Step 3.4.6

Simplify $\pm \sqrt{\frac{6}{1 - 7 x}}$ .

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Step 3.4.6.1

Rewrite $\sqrt{\frac{6}{1 - 7 x}}$ as $\frac{\sqrt{6}}{\sqrt{1 - 7 x}}$ .

$y = \pm \frac{\sqrt{6}}{\sqrt{1 - 7 x}}$

Step 3.4.6.2

Multiply $\frac{\sqrt{6}}{\sqrt{1 - 7 x}}$ by $\frac{\sqrt{1 - 7 x}}{\sqrt{1 - 7 x}}$ .

$y = \pm \frac{\sqrt{6}}{\sqrt{1 - 7 x}} \cdot \frac{\sqrt{1 - 7 x}}{\sqrt{1 - 7 x}}$

Step 3.4.6.3

Combine and simplify the denominator.

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Step 3.4.6.3.1

Multiply $\frac{\sqrt{6}}{\sqrt{1 - 7 x}}$ by $\frac{\sqrt{1 - 7 x}}{\sqrt{1 - 7 x}}$ .

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{\sqrt{1 - 7 x} \sqrt{1 - 7 x}}$

Step 3.4.6.3.2

Raise $\sqrt{1 - 7 x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{\sqrt{1 - 7 x}}^{1} \sqrt{1 - 7 x}}$

Step 3.4.6.3.3

Raise $\sqrt{1 - 7 x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{\sqrt{1 - 7 x}}^{1} {\sqrt{1 - 7 x}}^{1}}$

Step 3.4.6.3.4

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

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{\sqrt{1 - 7 x}}^{1 + 1}}$

Step 3.4.6.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{\sqrt{1 - 7 x}}^{2}}$

Step 3.4.6.3.6

Rewrite ${\sqrt{1 - 7 x}}^{2}$ as $1 - 7 x$ .

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Step 3.4.6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 7 x}$ as ${(1 - 7 x)}^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{({(1 - 7 x)}^{\frac{1}{2}})}^{2}}$

Step 3.4.6.3.6.2

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

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{(1 - 7 x)}^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3.6.3

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

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{(1 - 7 x)}^{\frac{2}{2}}}$

Step 3.4.6.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{(1 - 7 x)}^{\frac{2}{2}}}$

Step 3.4.6.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{{(1 - 7 x)}^{1}}$

Step 3.4.6.3.6.5

Simplify.

$y = \pm \frac{\sqrt{6} \sqrt{1 - 7 x}}{1 - 7 x}$

Step 3.4.6.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{6 (1 - 7 x)}}{1 - 7 x}$

Step 3.4.7

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

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Step 3.4.7.1

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