Algebra Examples

$\frac{2 x}{x + 7}$

Step 1

Interchange the variables.

$x = \frac{2 y}{y + 7}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{2 y}{y + 7} = x$ .

$\frac{2 y}{y + 7} = x$

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.

$y + 7, 1$

Step 2.2.2

Remove parentheses.

$y + 7, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$y + 7$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 y}{y + 7} (y + 7) = x (y + 7)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y + 7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 y}{y + 7} (y + 7) = x (y + 7)$

Step 2.3.2.1.2

Rewrite the expression.

$2 y = x (y + 7)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$2 y = x y + x \cdot 7$

Step 2.3.3.2

Move $7$ to the left of $x$ .

$2 y = x y + 7 x$

Step 2.4

Solve the equation.

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Step 2.4.1

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

$2 y - x y = 7 x$

Step 2.4.2

Factor $y$ out of $2 y - x y$ .

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Step 2.4.2.1

Factor $y$ out of $2 y$ .

$y \cdot 2 - x y = 7 x$

Step 2.4.2.2

Factor $y$ out of $- x y$ .

$y \cdot 2 + y (- x) = 7 x$

Step 2.4.2.3

Factor $y$ out of $y \cdot 2 + y (- x)$ .

$y (2 - x) = 7 x$

Step 2.4.3

Divide each term in $y (2 - x) = 7 x$ by $2 - x$ and simplify.

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Step 2.4.3.1

Divide each term in $y (2 - x) = 7 x$ by $2 - x$ .

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

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $2 - x$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{7 x}{2 - x}$ is the inverse of $f (x) = \frac{2 x}{x + 7}$ .

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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 $f^{- 1} (\frac{2 x}{x + 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{2 - (\frac{2 x}{x + 7})}$

Step 4.2.3

Combine $7$ and $\frac{2 x}{x + 7}$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{2 - \frac{2 x}{x + 7}}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 (x + 7)}{x + 7} - \frac{2 x}{x + 7}}$

Step 4.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 (x + 7) - 2 x}{x + 7}}$

Step 4.2.4.3

Reorder terms.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{- 2 x + 2 (x + 7)}{x + 7}}$

Step 4.2.4.4

Rewrite $\frac{- 2 x + 2 (x + 7)}{x + 7}$ in a factored form.

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Step 4.2.4.4.1

Factor $2$ out of $- 2 x + 2 (x + 7)$ .

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Step 4.2.4.4.1.1

Factor $2$ out of $- 2 x$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 (- x) + 2 (x + 7)}{x + 7}}$

Step 4.2.4.4.1.2

Factor $2$ out of $2 (- x) + 2 (x + 7)$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 (- x + x + 7)}{x + 7}}$

Step 4.2.4.4.2

Add $- x$ and $x$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 (0 + 7)}{x + 7}}$

Step 4.2.4.4.3

Add $0$ and $7$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{2 \cdot 7}{x + 7}}$

Step 4.2.4.5

Multiply $2$ by $7$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{14}{x + 7}}$

Step 4.2.5

Multiply $7$ by $2$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{\frac{14 x}{x + 7}}{\frac{14}{x + 7}}$

Step 4.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{14 x}{x + 7} \cdot \frac{x + 7}{14}$

Step 4.2.7

Cancel the common factor of $14$ .

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Step 4.2.7.1

Factor $14$ out of $14 x$ .

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{14 (x)}{x + 7} \cdot \frac{x + 7}{14}$

Step 4.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{14 x}{x + 7} \cdot \frac{x + 7}{14}$

Step 4.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{x}{x + 7} \cdot (x + 7)$

Step 4.2.8

Cancel the common factor of $x + 7$ .

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Step 4.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x + 7}) = \frac{x}{x + 7} \cdot (x + 7)$

Step 4.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x + 7}) = x$

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 $f (\frac{7 x}{2 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Combine $2$ and $\frac{7 x}{2 - x}$ .

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

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{2 - x}{2 - x}$ .

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

Step 4.3.4.2

Combine the numerators over the common denominator.

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

Step 4.3.4.3

Reorder terms.

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

Step 4.3.4.4

Rewrite $\frac{7 x + 7 (2 - x)}{- x + 2}$ in a factored form.

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Step 4.3.4.4.1

Factor $7$ out of $7 x + 7 (2 - x)$ .

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Step 4.3.4.4.1.1

Factor $7$ out of $7 x$ .

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

Step 4.3.4.4.1.2

Factor $7$ out of $7 (x) + 7 (2 - x)$ .

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

Step 4.3.4.4.2

Subtract $x$ from $x$ .

$f (\frac{7 x}{2 - x}) = \frac{\frac{2 (7 x)}{2 - x}}{\frac{7 (0 + 2)}{- x + 2}}$

Step 4.3.4.4.3

Add $0$ and $2$ .

$f (\frac{7 x}{2 - x}) = \frac{\frac{2 (7 x)}{2 - x}}{\frac{7 \cdot 2}{- x + 2}}$

Step 4.3.4.5

Multiply $7$ by $2$ .

$f (\frac{7 x}{2 - x}) = \frac{\frac{2 (7 x)}{2 - x}}{\frac{14}{- x + 2}}$

Step 4.3.5

Multiply $2$ by $7$ .

$f (\frac{7 x}{2 - x}) = \frac{\frac{14 x}{2 - x}}{\frac{14}{- x + 2}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x}{2 - x}) = \frac{14 x}{2 - x} \cdot \frac{- x + 2}{14}$

Step 4.3.7

Cancel the common factor of $14$ .

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Step 4.3.7.1

Factor $14$ out of $14 x$ .

$f (\frac{7 x}{2 - x}) = \frac{14 (x)}{2 - x} \cdot \frac{- x + 2}{14}$

Step 4.3.7.2

Cancel the common factor.

$f (\frac{7 x}{2 - x}) = \frac{14 x}{2 - x} \cdot \frac{- x + 2}{14}$

Step 4.3.7.3

Rewrite the expression.

$f (\frac{7 x}{2 - x}) = \frac{x}{2 - x} \cdot (- x + 2)$

Step 4.3.8

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

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

Step 4.3.9

Cancel the common factor of $- x + 2$ and $2 - x$ .

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Step 4.3.9.1

Reorder terms.

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

Step 4.3.9.2

Cancel the common factor.

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

Step 4.3.9.3

Divide $x$ by $1$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{2 - x}$ is the inverse of $f (x) = \frac{2 x}{x + 7}$ .

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