Algebra Examples

$\frac{6}{5 + x}$

Step 1

Interchange the variables.

$x = \frac{6}{5 + y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{6}{5 + y} = x$ .

$\frac{6}{5 + y} = x$

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.

$5 + y, 1$

Step 2.2.2

Remove parentheses.

$5 + y, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$5 + y$

Step 2.3

Multiply each term in $\frac{6}{5 + y} = x$ by $5 + y$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{6}{5 + y} = x$ by $5 + y$ .

$\frac{6}{5 + y} (5 + y) = x (5 + y)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5 + y$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6}{5 + y} (5 + y) = x (5 + y)$

Step 2.3.2.1.2

Rewrite the expression.

$6 = x (5 + y)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$6 = x \cdot 5 + x y$

Step 2.3.3.2

Move $5$ to the left of $x$ .

$6 = 5 x + x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $5 x + x y = 6$ .

$5 x + x y = 6$

Step 2.4.2

Subtract $5 x$ from both sides of the equation.

$x y = 6 - 5 x$

Step 2.4.3

Divide each term in $x y = 6 - 5 x$ by $x$ and simplify.

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Step 2.4.3.1

Divide each term in $x y = 6 - 5 x$ by $x$ .

$\frac{x y}{x} = \frac{6}{x} + \frac{- 5 x}{x}$

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.

$\frac{x y}{x} = \frac{6}{x} + \frac{- 5 x}{x}$

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{x} + \frac{- 5 x}{x}$

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.

$y = \frac{6}{x} + \frac{- 5 x}{x}$

Step 2.4.3.3.1.2

Divide $- 5$ by $1$ .

$y = \frac{6}{x} - 5$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{6}{x} - 5$

Step 4

Verify if $f^{- 1} (x) = \frac{6}{x} - 5$ is the inverse of $f (x) = \frac{6}{5 + x}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\frac{6}{5 + x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{5 + x}) = \frac{6}{\frac{6}{5 + x}} - 5$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6}{5 + x}) = 6 (\frac{5 + x}{6}) - 5$

Step 4.2.3.2

Cancel the common factor of $6$ .

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Step 4.2.3.2.1

Cancel the common factor.

$f^{- 1} (\frac{6}{5 + x}) = 6 (\frac{5 + x}{6}) - 5$

Step 4.2.3.2.2

Rewrite the expression.

$f^{- 1} (\frac{6}{5 + x}) = 5 + x - 5$

Step 4.2.4

Combine the opposite terms in $5 + x - 5$ .

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Step 4.2.4.1

Subtract $5$ from $5$ .

$f^{- 1} (\frac{6}{5 + x}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{6}{5 + x}) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{6}{x} - 5)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6}{x} - 5) = \frac{6}{5 + \frac{6}{x} - 5}$

Step 4.3.3

Remove parentheses.

$f (\frac{6}{x} - 5) = \frac{6}{5 + \frac{6}{x} - 5}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Subtract $5$ from $5$ .

$f (\frac{6}{x} - 5) = \frac{6}{\frac{6}{x} + 0}$

Step 4.3.4.2

Add $\frac{6}{x}$ and $0$ .

$f (\frac{6}{x} - 5) = \frac{6}{\frac{6}{x}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6}{x} - 5) = 6 (\frac{x}{6})$

Step 4.3.6

Cancel the common factor of $6$ .

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Step 4.3.6.1

Cancel the common factor.

$f (\frac{6}{x} - 5) = 6 (\frac{x}{6})$

Step 4.3.6.2

Rewrite the expression.

$f (\frac{6}{x} - 5) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6}{x} - 5$ is the inverse of $f (x) = \frac{6}{5 + x}$ .

$f^{- 1} (x) = \frac{6}{x} - 5$