Algebra Examples

$\frac{x - 6}{- 3 x}$

Step 1

Interchange the variables.

$x = \frac{y - 6}{- 3 y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{y - 6}{- 3 y} = x$ .

$\frac{y - 6}{- 3 y} = x$

Step 2.2

Move the negative in front of the fraction.

$- \frac{y - 6}{3 y} = x$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

$3 y, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$3 y$

Step 2.4

Multiply each term in $- \frac{y - 6}{3 y} = x$ by $3 y$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{y - 6}{3 y} = x$ by $3 y$ .

$- \frac{y - 6}{3 y} (3 y) = x (3 y)$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3 y$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{y - 6}{3 y}$ into the numerator.

$\frac{- (y - 6)}{3 y} (3 y) = x (3 y)$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- (y - 6)}{3 y} (3 y) = x (3 y)$

Step 2.4.2.1.3

Rewrite the expression.

$- (y - 6) = x (3 y)$

Step 2.4.2.2

Apply the distributive property.

$- y - - 6 = x (3 y)$

Step 2.4.2.3

Multiply $- 1$ by $- 6$ .

$- y + 6 = x (3 y)$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Rewrite using the commutative property of multiplication.

$- y + 6 = 3 x y$

Step 2.5

Solve the equation.

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Step 2.5.1

Subtract $3 x y$ from both sides of the equation.

$- y + 6 - 3 x y = 0$

Step 2.5.2

Subtract $6$ from both sides of the equation.

$- y - 3 x y = - 6$

Step 2.5.3

Factor $y$ out of $- y - 3 x y$ .

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Step 2.5.3.1

Factor $y$ out of $- y$ .

$y \cdot - 1 - 3 x y = - 6$

Step 2.5.3.2

Factor $y$ out of $- 3 x y$ .

$y \cdot - 1 + y (- 3 x) = - 6$

Step 2.5.3.3

Factor $y$ out of $y \cdot - 1 + y (- 3 x)$ .

$y (- 1 - 3 x) = - 6$

Step 2.5.4

Divide each term in $y (- 1 - 3 x) = - 6$ by $- 1 - 3 x$ and simplify.

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Step 2.5.4.1

Divide each term in $y (- 1 - 3 x) = - 6$ by $- 1 - 3 x$ .

$\frac{y (- 1 - 3 x)}{- 1 - 3 x} = \frac{- 6}{- 1 - 3 x}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $- 1 - 3 x$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{y (- 1 - 3 x)}{- 1 - 3 x} = \frac{- 6}{- 1 - 3 x}$

Step 2.5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 1 - 3 x}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Move the negative in front of the fraction.

$y = - \frac{6}{- 1 - 3 x}$

Step 2.5.4.3.2

Rewrite $- 1$ as $- 1 (1)$ .

$y = - \frac{6}{- 1 (1) - 3 x}$

Step 2.5.4.3.3

Factor $- 1$ out of $- 3 x$ .

$y = - \frac{6}{- 1 (1) - (3 x)}$

Step 2.5.4.3.4

Factor $- 1$ out of $- 1 (1) - (3 x)$ .

$y = - \frac{6}{- 1 (1 + 3 x)}$

Step 2.5.4.3.5

Simplify the expression.

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Step 2.5.4.3.5.1

Move the negative in front of the fraction.

$y = - - \frac{6}{1 + 3 x}$

Step 2.5.4.3.5.2

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{6}{1 + 3 x}$

Step 2.5.4.3.5.3

Multiply $\frac{6}{1 + 3 x}$ by $1$ .

$y = \frac{6}{1 + 3 x}$

Step 3

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

$f^{- 1} (x) = \frac{6}{1 + 3 x}$

Step 4

Verify if $f^{- 1} (x) = \frac{6}{1 + 3 x}$ is the inverse of $f (x) = \frac{x - 6}{- 3 x}$ .

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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 $f^{- 1} (\frac{x - 6}{- 3 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{1 + 3 (\frac{x - 6}{- 3 x})}$

Step 4.2.3

Simplify the denominator.

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Step 4.2.3.1

Cancel the common factor of $3$ .

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Step 4.2.3.1.1

Factor $3$ out of $- 3 x$ .

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{1 + 3 (\frac{x - 6}{3 (- x)})}$

Step 4.2.3.1.2

Cancel the common factor.

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{1 + 3 (\frac{x - 6}{3 (- x)})}$

Step 4.2.3.1.3

Rewrite the expression.

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{1 + \frac{x - 6}{- x}}$

Step 4.2.3.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{1 - \frac{x - 6}{x}}$

Step 4.2.3.3

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

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{x}{x} - \frac{x - 6}{x}}$

Step 4.2.3.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{x - (x - 6)}{x}}$

Step 4.2.3.5

Rewrite $\frac{x - (x - 6)}{x}$ in a factored form.

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Step 4.2.3.5.1

Apply the distributive property.

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{x - x + 6}{x}}$

Step 4.2.3.5.2

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{x - x + 6}{x}}$

Step 4.2.3.5.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{0 + 6}{x}}$

Step 4.2.3.5.4

Add $0$ and $6$ .

$f^{- 1} (\frac{x - 6}{- 3 x}) = \frac{6}{\frac{6}{x}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x - 6}{- 3 x}) = 6 (\frac{x}{6})$

Step 4.2.5

Cancel the common factor of $6$ .

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Step 4.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{x - 6}{- 3 x}) = 6 (\frac{x}{6})$

Step 4.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{x - 6}{- 3 x}) = x$

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 $f (\frac{6}{1 + 3 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6}{1 + 3 x}) = \frac{(\frac{6}{1 + 3 x}) - 6}{- 3 \frac{6}{1 + 3 x}}$

Step 4.3.3

Cancel the common factor of $(\frac{6}{1 + 3 x}) - 6$ and $- 3$ .

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Step 4.3.3.1

Factor $3$ out of $\frac{6}{1 + 3 x}$ .

$f (\frac{6}{1 + 3 x}) = \frac{3 (\frac{2}{1 + 3 x}) - 6}{- 3 \frac{6}{1 + 3 x}}$

Step 4.3.3.2

Factor $3$ out of $- 6$ .

$f (\frac{6}{1 + 3 x}) = \frac{3 (\frac{2}{1 + 3 x}) + 3 \cdot - 2}{- 3 \frac{6}{1 + 3 x}}$

Step 4.3.3.3

Factor $3$ out of $3 \frac{2}{1 + 3 x} + 3 \cdot - 2$ .

$f (\frac{6}{1 + 3 x}) = \frac{3 (\frac{2}{1 + 3 x} - 2)}{- 3 \frac{6}{1 + 3 x}}$

Step 4.3.3.4

Cancel the common factors.

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Step 4.3.3.4.1

Factor $3$ out of $- 3 \frac{6}{1 + 3 x}$ .

$f (\frac{6}{1 + 3 x}) = \frac{3 (\frac{2}{1 + 3 x} - 2)}{3 (- \frac{6}{1 + 3 x})}$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{6}{1 + 3 x}) = \frac{3 (\frac{2}{1 + 3 x} - 2)}{3 (- \frac{6}{1 + 3 x})}$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{6}{1 + 3 x}) = \frac{\frac{2}{1 + 3 x} - 2}{- \frac{6}{1 + 3 x}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6}{1 + 3 x}) = (\frac{2}{1 + 3 x} - 2) (- \frac{1 + 3 x}{6})$

Step 4.3.5

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{1 + 3 x}{1 + 3 x}$ .

$f (\frac{6}{1 + 3 x}) = (\frac{2}{1 + 3 x} - 2 \cdot \frac{1 + 3 x}{1 + 3 x}) (- \frac{1 + 3 x}{6})$

Step 4.3.6

Simplify terms.

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Step 4.3.6.1

Combine $- 2$ and $\frac{1 + 3 x}{1 + 3 x}$ .

$f (\frac{6}{1 + 3 x}) = (\frac{2}{1 + 3 x} + \frac{- 2 (1 + 3 x)}{1 + 3 x}) (- \frac{1 + 3 x}{6})$

Step 4.3.6.2

Combine the numerators over the common denominator.

$f (\frac{6}{1 + 3 x}) = \frac{2 - 2 (1 + 3 x)}{1 + 3 x} \cdot (- \frac{1 + 3 x}{6})$

Step 4.3.6.3

Rewrite using the commutative property of multiplication.

$f (\frac{6}{1 + 3 x}) = - \frac{2 - 2 (1 + 3 x)}{1 + 3 x} \cdot \frac{1 + 3 x}{6}$

Step 4.3.6.4

Cancel the common factor of $1 + 3 x$ .

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Step 4.3.6.4.1

Move the leading negative in $- \frac{2 - 2 (1 + 3 x)}{1 + 3 x}$ into the numerator.

$f (\frac{6}{1 + 3 x}) = \frac{- (2 - 2 (1 + 3 x))}{1 + 3 x} \cdot \frac{1 + 3 x}{6}$

Step 4.3.6.4.2

Cancel the common factor.

$f (\frac{6}{1 + 3 x}) = \frac{- (2 - 2 (1 + 3 x))}{1 + 3 x} \cdot \frac{1 + 3 x}{6}$

Step 4.3.6.4.3

Rewrite the expression.

$f (\frac{6}{1 + 3 x}) = - (2 - 2 (1 + 3 x)) \frac{1}{6}$

Step 4.3.7

Simplify each term.

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Step 4.3.7.1

Apply the distributive property.

$f (\frac{6}{1 + 3 x}) = - (2 - 2 \cdot 1 - 2 (3 x)) \frac{1}{6}$

Step 4.3.7.2

Multiply $- 2$ by $1$ .

$f (\frac{6}{1 + 3 x}) = - (2 - 2 - 2 (3 x)) \frac{1}{6}$

Step 4.3.7.3

Multiply $3$ by $- 2$ .

$f (\frac{6}{1 + 3 x}) = - (2 - 2 - 6 x) \frac{1}{6}$

Step 4.3.8

Reduce the expression by cancelling the common factors.

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Step 4.3.8.1

Subtract $2$ from $2$ .

$f (\frac{6}{1 + 3 x}) = - (0 - 6 x) \frac{1}{6}$

Step 4.3.8.2

Subtract $6 x$ from $0$ .

$f (\frac{6}{1 + 3 x}) = 6 x \frac{1}{6}$

Step 4.3.8.3

Cancel the common factor of $6$ .

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Step 4.3.8.3.1

Factor $6$ out of $- (- 6 x)$ .

$f (\frac{6}{1 + 3 x}) = 6 (x) (\frac{1}{6})$

Step 4.3.8.3.2

Cancel the common factor.

$f (\frac{6}{1 + 3 x}) = 6 (x) (\frac{1}{6})$

Step 4.3.8.3.3

Rewrite the expression.

$f (\frac{6}{1 + 3 x}) = x$

Step 4.3.8.4

Multiply.

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Step 4.3.8.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{6}{1 + 3 x}) = 1 x$

Step 4.3.8.4.2

Multiply $x$ by $1$ .

$f (\frac{6}{1 + 3 x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6}{1 + 3 x}$ is the inverse of $f (x) = \frac{x - 6}{- 3 x}$ .

$f^{- 1} (x) = \frac{6}{1 + 3 x}$