Algebra Examples

$y = \frac{x - 4}{33 - x}$

Step 1

Interchange the variables.

$x = \frac{y - 4}{33 - y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{y - 4}{33 - y} = x$ .

$\frac{y - 4}{33 - y} = x$

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.

$33 - y, 1$

Step 2.2.2

Remove parentheses.

$33 - y, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$33 - y$

Step 2.3

Multiply each term in $\frac{y - 4}{33 - y} = x$ by $33 - y$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{y - 4}{33 - y} = x$ by $33 - y$ .

$\frac{y - 4}{33 - y} (33 - y) = x (33 - y)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $33 - y$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{y - 4}{33 - y} (33 - y) = x (33 - y)$

Step 2.3.2.1.2

Rewrite the expression.

$y - 4 = x (33 - y)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$y - 4 = x \cdot 33 + x (- y)$

Step 2.3.3.2

Reorder.

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Step 2.3.3.2.1

Move $33$ to the left of $x$ .

$y - 4 = 33 \cdot x + x (- y)$

Step 2.3.3.2.2

Rewrite using the commutative property of multiplication.

$y - 4 = 33 x - x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $x y$ to both sides of the equation.

$y - 4 + x y = 33 x$

Step 2.4.2

Add $4$ to both sides of the equation.

$y + x y = 33 x + 4$

Step 2.4.3

Factor $y$ out of $y + x y$ .

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Step 2.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 + x y = 33 x + 4$

Step 2.4.3.2

Factor $y$ out of $x y$ .

$y \cdot 1 + y x = 33 x + 4$

Step 2.4.3.3

Factor $y$ out of $y \cdot 1 + y x$ .

$y (1 + x) = 33 x + 4$

Step 2.4.4

Divide each term in $y (1 + x) = 33 x + 4$ by $1 + x$ and simplify.

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Step 2.4.4.1

Divide each term in $y (1 + x) = 33 x + 4$ by $1 + x$ .

$\frac{y (1 + x)}{1 + x} = \frac{33 x}{1 + x} + \frac{4}{1 + x}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $1 + x$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (1 + x)}{1 + x} = \frac{33 x}{1 + x} + \frac{4}{1 + x}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{33 x}{1 + x} + \frac{4}{1 + x}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{33 x + 4}{1 + x}$

Step 3

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

$f^{- 1} (x) = \frac{33 x + 4}{1 + x}$

Step 4

Verify if $f^{- 1} (x) = \frac{33 x + 4}{1 + x}$ is the inverse of $f (x) = \frac{x - 4}{33 - x}$ .

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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 $f^{- 1} (\frac{x - 4}{33 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{33 (\frac{x - 4}{33 - x}) + 4}{1 + \frac{x - 4}{33 - x}}$

Step 4.2.3

Remove parentheses.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{33 (\frac{x - 4}{33 - x}) + 4}{1 + \frac{x - 4}{33 - x}}$

Step 4.2.4

Multiply the numerator and denominator of the fraction by $33 - x$ .

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Step 4.2.4.1

Multiply $\frac{33 \frac{x - 4}{33 - x} + 4}{1 + \frac{x - 4}{33 - x}}$ by $\frac{33 - x}{33 - x}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{33 - x}{33 - x} \cdot \frac{33 (\frac{x - 4}{33 - x}) + 4}{1 + \frac{x - 4}{33 - x}}$

Step 4.2.4.2

Combine.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x}) + 4)}{(33 - x) (1 + \frac{x - 4}{33 - x})}$

Step 4.2.5

Apply the distributive property.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x})) + (33 - x) \cdot 4}{(33 - x) \cdot 1 + (33 - x) (\frac{x - 4}{33 - x})}$

Step 4.2.6

Cancel the common factor of $33 - x$ .

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Step 4.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x})) + (33 - x) \cdot 4}{(33 - x) \cdot 1 + (33 - x) (\frac{x - 4}{33 - x})}$

Step 4.2.6.2

Rewrite the expression.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x})) + (33 - x) \cdot 4}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7

Simplify the numerator.

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Step 4.2.7.1

Factor $33 - x$ out of $(33 - x) \cdot 33 \frac{x - 4}{33 - x} + (33 - x) \cdot 4$ .

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Step 4.2.7.1.1

Factor $33 - x$ out of $(33 - x) \cdot 33 \frac{x - 4}{33 - x}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x})) + (33 - x) \cdot 4}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.1.2

Factor $33 - x$ out of $(33 - x) \cdot 4$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x})) + (33 - x) (4)}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.1.3

Factor $33 - x$ out of $(33 - x) (33 \frac{x - 4}{33 - x}) + (33 - x) (4)$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (33 (\frac{x - 4}{33 - x}) + 4)}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.2

Combine $33$ and $\frac{x - 4}{33 - x}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 (x - 4)}{33 - x} + 4)}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{33 - x}{33 - x}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 (x - 4)}{33 - x} + \frac{4 (33 - x)}{33 - x})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 (x - 4) + 4 (33 - x)}{33 - x})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.5

Reorder terms.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 (x - 4) + 4 (33 - x)}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6

Rewrite $\frac{33 (x - 4) + 4 (33 - x)}{- x + 33}$ in a factored form.

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Step 4.2.7.6.1

Apply the distributive property.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 x + 33 \cdot - 4 + 4 (33 - x)}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.2

Multiply $33$ by $- 4$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 x - 132 + 4 (33 - x)}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.3

Apply the distributive property.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 x - 132 + 4 \cdot 33 + 4 (- x)}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.4

Multiply $4$ by $33$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 x - 132 + 132 + 4 (- x)}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.5

Multiply $- 1$ by $4$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{33 x - 132 + 132 - 4 x}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.6

Subtract $4 x$ from $33 x$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x - 132 + 132}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.7

Add $- 132$ and $132$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x + 0}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.7.6.8

Add $29 x$ and $0$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x}{- x + 33})}{(33 - x) \cdot 1 + x - 4}$

Step 4.2.8

Simplify the denominator.

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Step 4.2.8.1

Multiply $33 - x$ by $1$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x}{- x + 33})}{33 - x + x - 4}$

Step 4.2.8.2

Subtract $4$ from $33$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x}{- x + 33})}{- x + x + 29}$

Step 4.2.8.3

Add $- x$ and $x$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x}{- x + 33})}{0 + 29}$

Step 4.2.8.4

Add $0$ and $29$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{(33 - x) (\frac{29 x}{- x + 33})}{29}$

Step 4.2.9

Simplify terms.

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Step 4.2.9.1

Factor $\frac{29 x}{- x + 33}$ out of $\frac{(33 - x) \frac{29 x}{- x + 33}}{29}$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{29 x}{- x + 33} \cdot \frac{33 - x}{29}$

Step 4.2.9.2

Cancel the common factor of $29$ .

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Step 4.2.9.2.1

Factor $29$ out of $29 x$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{29 (x)}{- x + 33} \cdot \frac{33 - x}{29}$

Step 4.2.9.2.2

Cancel the common factor.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{29 x}{- x + 33} \cdot \frac{33 - x}{29}$

Step 4.2.9.2.3

Rewrite the expression.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{x}{- x + 33} \cdot (33 - x)$

Step 4.2.9.3

Multiply $\frac{x}{- x + 33}$ by $33 - x$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{x (33 - x)}{- x + 33}$

Step 4.2.9.4

Cancel the common factor of $33 - x$ and $- x + 33$ .

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Step 4.2.9.4.1

Reorder terms.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{x (- x + 33)}{- x + 33}$

Step 4.2.9.4.2

Cancel the common factor.

$f^{- 1} (\frac{x - 4}{33 - x}) = \frac{x (- x + 33)}{- x + 33}$

Step 4.2.9.4.3

Divide $x$ by $1$ .

$f^{- 1} (\frac{x - 4}{33 - x}) = x$

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 $f (\frac{33 x + 4}{1 + x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{(\frac{33 x + 4}{1 + x}) - 4}{33 - (\frac{33 x + 4}{1 + x})}$

Step 4.3.3

Multiply the numerator and denominator of the fraction by $1 + x$ .

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Step 4.3.3.1

Multiply $\frac{\frac{33 x + 4}{1 + x} - 4}{33 - \frac{33 x + 4}{1 + x}}$ by $\frac{1 + x}{1 + x}$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{1 + x}{1 + x} \cdot \frac{\frac{33 x + 4}{1 + x} - 4}{33 - \frac{33 x + 4}{1 + x}}$

Step 4.3.3.2

Combine.

$f (\frac{33 x + 4}{1 + x}) = \frac{(1 + x) (\frac{33 x + 4}{1 + x} - 4)}{(1 + x) (33 - \frac{33 x + 4}{1 + x})}$

Step 4.3.4

Apply the distributive property.

$f (\frac{33 x + 4}{1 + x}) = \frac{(1 + x) (\frac{33 x + 4}{1 + x}) + (1 + x) \cdot - 4}{(1 + x) \cdot 33 + (1 + x) (- \frac{33 x + 4}{1 + x})}$

Step 4.3.5

Simplify by cancelling.

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Step 4.3.5.1

Cancel the common factor of $1 + x$ .

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Step 4.3.5.1.1

Cancel the common factor.

$f (\frac{33 x + 4}{1 + x}) = \frac{(1 + x) (\frac{33 x + 4}{1 + x}) + (1 + x) \cdot - 4}{(1 + x) \cdot 33 + (1 + x) (- \frac{33 x + 4}{1 + x})}$

Step 4.3.5.1.2

Rewrite the expression.

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 + (1 + x) \cdot - 4}{(1 + x) \cdot 33 + (1 + x) (- \frac{33 x + 4}{1 + x})}$

Step 4.3.5.2

Cancel the common factor of $1 + x$ .

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Step 4.3.5.2.1

Move the leading negative in $- \frac{33 x + 4}{1 + x}$ into the numerator.

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 + (1 + x) \cdot - 4}{(1 + x) \cdot 33 + (1 + x) (\frac{- (33 x + 4)}{1 + x})}$

Step 4.3.5.2.2

Cancel the common factor.

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 + (1 + x) \cdot - 4}{(1 + x) \cdot 33 + (1 + x) (\frac{- (33 x + 4)}{1 + x})}$

Step 4.3.5.2.3

Rewrite the expression.

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 + (1 + x) \cdot - 4}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Apply the distributive property.

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 + 1 \cdot - 4 + x \cdot - 4}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6.2

Multiply $- 4$ by $1$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 - 4 + x \cdot - 4}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6.3

Move $- 4$ to the left of $x$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{33 x + 4 - 4 - 4 \cdot x}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6.4

Subtract $4 x$ from $33 x$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x + 4 - 4}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6.5

Subtract $4$ from $4$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x + 0}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.6.6

Add $29 x$ and $0$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{(1 + x) \cdot 33 - (33 x + 4)}$

Step 4.3.7

Simplify the denominator.

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Step 4.3.7.1

Apply the distributive property.

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{1 \cdot 33 + x \cdot 33 - (33 x + 4)}$

Step 4.3.7.2

Multiply $33$ by $1$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 + x \cdot 33 - (33 x + 4)}$

Step 4.3.7.3

Move $33$ to the left of $x$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 + 33 \cdot x - (33 x + 4)}$

Step 4.3.7.4

Apply the distributive property.

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 + 33 x - (33 x) - 1 \cdot 4}$

Step 4.3.7.5

Multiply $33$ by $- 1$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 + 33 x - 33 x - 1 \cdot 4}$

Step 4.3.7.6

Multiply $- 1$ by $4$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 + 33 x - 33 x - 4}$

Step 4.3.7.7

Subtract $4$ from $33$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{33 x - 33 x + 29}$

Step 4.3.7.8

Subtract $33 x$ from $33 x$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{0 + 29}$

Step 4.3.7.9

Add $0$ and $29$ .

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{29}$

Step 4.3.8

Cancel the common factor of $29$ .

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Step 4.3.8.1

Cancel the common factor.

$f (\frac{33 x + 4}{1 + x}) = \frac{29 x}{29}$

Step 4.3.8.2

Divide $x$ by $1$ .

$f (\frac{33 x + 4}{1 + x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{33 x + 4}{1 + x}$ is the inverse of $f (x) = \frac{x - 4}{33 - x}$ .

$f^{- 1} (x) = \frac{33 x + 4}{1 + x}$