Algebra Examples

$f (x) = \frac{7 x - 3}{6 x - 7}$

Step 1

Write $f (x) = \frac{7 x - 3}{6 x - 7}$ as an equation.

$y = \frac{7 x - 3}{6 x - 7}$

Step 2

Interchange the variables.

$x = \frac{7 y - 3}{6 y - 7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{7 y - 3}{6 y - 7} = x$ .

$\frac{7 y - 3}{6 y - 7} = x$

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.

$6 y - 7, 1$

Step 3.2.2

Remove parentheses.

$6 y - 7, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$6 y - 7$

Step 3.3

Multiply each term in $\frac{7 y - 3}{6 y - 7} = x$ by $6 y - 7$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{7 y - 3}{6 y - 7} = x$ by $6 y - 7$ .

$\frac{7 y - 3}{6 y - 7} (6 y - 7) = x (6 y - 7)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6 y - 7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{7 y - 3}{6 y - 7} (6 y - 7) = x (6 y - 7)$

Step 3.3.2.1.2

Rewrite the expression.

$7 y - 3 = x (6 y - 7)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$7 y - 3 = x (6 y) + x \cdot - 7$

Step 3.3.3.2

Reorder.

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Step 3.3.3.2.1

Rewrite using the commutative property of multiplication.

$7 y - 3 = 6 x y + x \cdot - 7$

Step 3.3.3.2.2

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

$7 y - 3 = 6 x y - 7 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Subtract $6 x y$ from both sides of the equation.

$7 y - 3 - 6 x y = - 7 x$

Step 3.4.2

Add $3$ to both sides of the equation.

$7 y - 6 x y = - 7 x + 3$

Step 3.4.3

Factor $y$ out of $7 y - 6 x y$ .

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Step 3.4.3.1

Factor $y$ out of $7 y$ .

$y \cdot 7 - 6 x y = - 7 x + 3$

Step 3.4.3.2

Factor $y$ out of $- 6 x y$ .

$y \cdot 7 + y (- 6 x) = - 7 x + 3$

Step 3.4.3.3

Factor $y$ out of $y \cdot 7 + y (- 6 x)$ .

$y (7 - 6 x) = - 7 x + 3$

Step 3.4.4

Divide each term in $y (7 - 6 x) = - 7 x + 3$ by $7 - 6 x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (7 - 6 x) = - 7 x + 3$ by $7 - 6 x$ .

$\frac{y (7 - 6 x)}{7 - 6 x} = \frac{- 7 x}{7 - 6 x} + \frac{3}{7 - 6 x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $7 - 6 x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (7 - 6 x)}{7 - 6 x} = \frac{- 7 x}{7 - 6 x} + \frac{3}{7 - 6 x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7 x}{7 - 6 x} + \frac{3}{7 - 6 x}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- 7 x + 3}{7 - 6 x}$

Step 3.4.4.3.2

Factor $- 1$ out of $- 7 x$ .

$y = \frac{- (7 x) + 3}{7 - 6 x}$

Step 3.4.4.3.3

Rewrite $3$ as $- 1 (- 3)$ .

$y = \frac{- (7 x) - 1 (- 3)}{7 - 6 x}$

Step 3.4.4.3.4

Factor $- 1$ out of $- (7 x) - 1 (- 3)$ .

$y = \frac{- (7 x - 3)}{7 - 6 x}$

Step 3.4.4.3.5

Simplify the expression.

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Step 3.4.4.3.5.1

Rewrite $- (7 x - 3)$ as $- 1 (7 x - 3)$ .

$y = \frac{- 1 (7 x - 3)}{7 - 6 x}$

Step 3.4.4.3.5.2

Move the negative in front of the fraction.

$y = - \frac{7 x - 3}{7 - 6 x}$

Step 4

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

$f^{- 1} (x) = - \frac{7 x - 3}{7 - 6 x}$

Step 5

Verify if $f^{- 1} (x) = - \frac{7 x - 3}{7 - 6 x}$ is the inverse of $f (x) = \frac{7 x - 3}{6 x - 7}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{7 x - 3}{6 x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{7 (\frac{7 x - 3}{6 x - 7}) - 3}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Combine $7$ and $\frac{7 x - 3}{6 x - 7}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{7 (7 x - 3)}{6 x - 7} - 3}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{6 x - 7}{6 x - 7}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{7 (7 x - 3)}{6 x - 7} - 3 \cdot \frac{6 x - 7}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.3

Combine $- 3$ and $\frac{6 x - 7}{6 x - 7}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{7 (7 x - 3)}{6 x - 7} + \frac{- 3 (6 x - 7)}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{7 (7 x - 3) - 3 (6 x - 7)}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5

Rewrite $\frac{7 (7 x - 3) - 3 (6 x - 7)}{6 x - 7}$ in a factored form.

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Step 5.2.3.5.1

Apply the distributive property.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{7 (7 x) + 7 \cdot - 3 - 3 (6 x - 7)}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.2

Multiply $7$ by $7$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{49 x + 7 \cdot - 3 - 3 (6 x - 7)}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.3

Multiply $7$ by $- 3$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{49 x - 21 - 3 (6 x - 7)}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.4

Apply the distributive property.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{49 x - 21 - 3 (6 x) - 3 \cdot - 7}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.5

Multiply $6$ by $- 3$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{49 x - 21 - 18 x - 3 \cdot - 7}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.6

Multiply $- 3$ by $- 7$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{49 x - 21 - 18 x + 21}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.7

Subtract $18 x$ from $49 x$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x - 21 + 21}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.8

Add $- 21$ and $21$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x + 0}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.3.5.9

Add $31 x$ and $0$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{7 - 6 \frac{7 x - 3}{6 x - 7}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Combine $- 6$ and $\frac{7 x - 3}{6 x - 7}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{7 + \frac{- 6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{7 - \frac{6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.3

To write $7$ as a fraction with a common denominator, multiply by $\frac{6 x - 7}{6 x - 7}$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{7 (6 x - 7)}{6 x - 7} - \frac{6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{7 (6 x - 7) - 6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.5

Rewrite $\frac{7 (6 x - 7) - 6 (7 x - 3)}{6 x - 7}$ in a factored form.

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Step 5.2.4.5.1

Apply the distributive property.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{7 (6 x) + 7 \cdot - 7 - 6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.5.2

Multiply $6$ by $7$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{42 x + 7 \cdot - 7 - 6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.5.3

Multiply $7$ by $- 7$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{42 x - 49 - 6 (7 x - 3)}{6 x - 7}}$

Step 5.2.4.5.4

Apply the distributive property.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{42 x - 49 - 6 (7 x) - 6 \cdot - 3}{6 x - 7}}$

Step 5.2.4.5.5

Multiply $7$ by $- 6$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{42 x - 49 - 42 x - 6 \cdot - 3}{6 x - 7}}$

Step 5.2.4.5.6

Multiply $- 6$ by $- 3$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{42 x - 49 - 42 x + 18}{6 x - 7}}$

Step 5.2.4.5.7

Subtract $42 x$ from $42 x$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{0 - 49 + 18}{6 x - 7}}$

Step 5.2.4.5.8

Subtract $49$ from $0$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{- 49 + 18}{6 x - 7}}$

Step 5.2.4.5.9

Add $- 49$ and $18$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{\frac{- 31}{6 x - 7}}$

Step 5.2.4.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - \frac{\frac{31 x}{6 x - 7}}{- \frac{31}{6 x - 7}}$

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{31 x}{6 x - 7} \cdot (- \frac{6 x - 7}{31}))$

Step 5.2.6

Rewrite using the commutative property of multiplication.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (- \frac{31 x}{6 x - 7} \cdot \frac{6 x - 7}{31})$

Step 5.2.7

Cancel the common factor of $31$ .

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Step 5.2.7.1

Move the leading negative in $- \frac{31 x}{6 x - 7}$ into the numerator.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{- 31 x}{6 x - 7} \cdot \frac{6 x - 7}{31})$

Step 5.2.7.2

Factor $31$ out of $- 31 x$ .

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{31 (- x)}{6 x - 7} \cdot \frac{6 x - 7}{31})$

Step 5.2.7.3

Cancel the common factor.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{31 (- x)}{6 x - 7} \cdot \frac{6 x - 7}{31})$

Step 5.2.7.4

Rewrite the expression.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{- x}{6 x - 7} \cdot (6 x - 7))$

Step 5.2.8

Cancel the common factor of $6 x - 7$ .

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = - (\frac{- x}{6 x - 7} \cdot (6 x - 7))$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{7 x - 3}{6 x - 7}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (- \frac{7 x - 3}{7 - 6 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{7 (- \frac{7 x - 3}{7 - 6 x}) - 3}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Multiply $7 (- \frac{7 x - 3}{7 - 6 x})$ .

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Step 5.3.3.1.1

Multiply $- 1$ by $7$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- 7 \frac{7 x - 3}{7 - 6 x} - 3}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.1.2

Combine $- 7$ and $\frac{7 x - 3}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- 7 (7 x - 3)}{7 - 6 x} - 3}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.2

Move the negative in front of the fraction.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{7 (7 x - 3)}{7 - 6 x} - 3}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{7 - 6 x}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{7 (7 x - 3)}{7 - 6 x} - 3 \cdot \frac{7 - 6 x}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.4

Combine $- 3$ and $\frac{7 - 6 x}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{7 (7 x - 3)}{7 - 6 x} + \frac{- 3 (7 - 6 x)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.5

Combine the numerators over the common denominator.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- 7 (7 x - 3) - 3 (7 - 6 x)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6

Rewrite $\frac{- 7 (7 x - 3) - 3 (7 - 6 x)}{7 - 6 x}$ in a factored form.

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Step 5.3.3.6.1

Factor $- 1$ out of $- 7 (7 x - 3) - 3 (7 - 6 x)$ .

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Step 5.3.3.6.1.1

Reorder $7$ and $- 6 x$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- 7 (7 x - 3) - 3 (- 6 x + 7)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.1.2

Factor $- 1$ out of $- 7 (7 x - 3)$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (7 (7 x - 3)) - 3 (- 6 x + 7)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.1.3

Factor $- 1$ out of $- 3 (- 6 x + 7)$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (7 (7 x - 3)) - (3 (- 6 x + 7))}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.1.4

Factor $- 1$ out of $- (7 (7 x - 3)) - (3 (- 6 x + 7))$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (7 (7 x - 3) + 3 (- 6 x + 7))}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.2

Apply the distributive property.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (7 (7 x) + 7 \cdot - 3 + 3 (- 6 x + 7))}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.3

Multiply $7$ by $7$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (49 x + 7 \cdot - 3 + 3 (- 6 x + 7))}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.4

Multiply $7$ by $- 3$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (49 x - 21 + 3 (- 6 x + 7))}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.5

Apply the distributive property.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (49 x - 21 + 3 (- 6 x) + 3 \cdot 7)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.6

Multiply $- 6$ by $3$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (49 x - 21 - 18 x + 3 \cdot 7)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.7

Multiply $3$ by $7$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (49 x - 21 - 18 x + 21)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.8

Subtract $18 x$ from $49 x$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (31 x - 21 + 21)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.9

Add $- 21$ and $21$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- (31 x + 0)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.10

Add $31 x$ and $0$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- 1 \cdot (31 x)}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.6.11

Multiply $- 1$ by $31$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{- 31 x}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.3.7

Move the negative in front of the fraction.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{6 (- \frac{7 x - 3}{7 - 6 x}) - 7}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $6 (- \frac{7 x - 3}{7 - 6 x})$ .

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Step 5.3.4.1.1

Multiply $- 1$ by $6$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{- 6 \frac{7 x - 3}{7 - 6 x} - 7}$

Step 5.3.4.1.2

Combine $- 6$ and $\frac{7 x - 3}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 6 (7 x - 3)}{7 - 6 x} - 7}$

Step 5.3.4.2

Move the negative in front of the fraction.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{- \frac{6 (7 x - 3)}{7 - 6 x} - 7}$

Step 5.3.4.3

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{7 - 6 x}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{- \frac{6 (7 x - 3)}{7 - 6 x} - 7 \cdot \frac{7 - 6 x}{7 - 6 x}}$

Step 5.3.4.4

Combine $- 7$ and $\frac{7 - 6 x}{7 - 6 x}$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{- \frac{6 (7 x - 3)}{7 - 6 x} + \frac{- 7 (7 - 6 x)}{7 - 6 x}}$

Step 5.3.4.5

Combine the numerators over the common denominator.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 6 (7 x - 3) - 7 (7 - 6 x)}{7 - 6 x}}$

Step 5.3.4.6

Reorder terms.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 6 (7 x - 3) - 7 (7 - 6 x)}{- 6 x + 7}}$

Step 5.3.4.7

Rewrite $\frac{- 6 (7 x - 3) - 7 (7 - 6 x)}{- 6 x + 7}$ in a factored form.

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Step 5.3.4.7.1

Factor $- 1$ out of $- 6 (7 x - 3) - 7 (7 - 6 x)$ .

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Step 5.3.4.7.1.1

Reorder $7$ and $- 6 x$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 6 (7 x - 3) - 7 (- 6 x + 7)}{- 6 x + 7}}$

Step 5.3.4.7.1.2

Factor $- 1$ out of $- 6 (7 x - 3)$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (6 (7 x - 3)) - 7 (- 6 x + 7)}{- 6 x + 7}}$

Step 5.3.4.7.1.3

Factor $- 1$ out of $- 7 (- 6 x + 7)$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (6 (7 x - 3)) - (7 (- 6 x + 7))}{- 6 x + 7}}$

Step 5.3.4.7.1.4

Factor $- 1$ out of $- (6 (7 x - 3)) - (7 (- 6 x + 7))$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (6 (7 x - 3) + 7 (- 6 x + 7))}{- 6 x + 7}}$

Step 5.3.4.7.2

Apply the distributive property.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (6 (7 x) + 6 \cdot - 3 + 7 (- 6 x + 7))}{- 6 x + 7}}$

Step 5.3.4.7.3

Multiply $7$ by $6$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (42 x + 6 \cdot - 3 + 7 (- 6 x + 7))}{- 6 x + 7}}$

Step 5.3.4.7.4

Multiply $6$ by $- 3$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (42 x - 18 + 7 (- 6 x + 7))}{- 6 x + 7}}$

Step 5.3.4.7.5

Apply the distributive property.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (42 x - 18 + 7 (- 6 x) + 7 \cdot 7)}{- 6 x + 7}}$

Step 5.3.4.7.6

Multiply $- 6$ by $7$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (42 x - 18 - 42 x + 7 \cdot 7)}{- 6 x + 7}}$

Step 5.3.4.7.7

Multiply $7$ by $7$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (42 x - 18 - 42 x + 49)}{- 6 x + 7}}$

Step 5.3.4.7.8

Subtract $42 x$ from $42 x$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (0 - 18 + 49)}{- 6 x + 7}}$

Step 5.3.4.7.9

Subtract $18$ from $0$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- (- 18 + 49)}{- 6 x + 7}}$

Step 5.3.4.7.10

Add $- 18$ and $49$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 1 \cdot 31}{- 6 x + 7}}$

Step 5.3.4.8

Multiply $- 1$ by $31$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{\frac{- 31}{- 6 x + 7}}$

Step 5.3.4.9

Move the negative in front of the fraction.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{- \frac{31 x}{7 - 6 x}}{- \frac{31}{- 6 x + 7}}$

Step 5.3.5

Dividing two negative values results in a positive value.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{\frac{31 x}{7 - 6 x}}{\frac{31}{- 6 x + 7}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{31 x}{7 - 6 x} \cdot \frac{- 6 x + 7}{31}$

Step 5.3.7

Cancel the common factor of $31$ .

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Step 5.3.7.1

Factor $31$ out of $31 x$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{31 (x)}{7 - 6 x} \cdot \frac{- 6 x + 7}{31}$

Step 5.3.7.2

Cancel the common factor.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{31 x}{7 - 6 x} \cdot \frac{- 6 x + 7}{31}$

Step 5.3.7.3

Rewrite the expression.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{x}{7 - 6 x} \cdot (- 6 x + 7)$

Step 5.3.8

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

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{x (- 6 x + 7)}{7 - 6 x}$

Step 5.3.9

Cancel the common factor of $- 6 x + 7$ and $7 - 6 x$ .

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Step 5.3.9.1

Reorder terms.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{x (- 6 x + 7)}{- 6 x + 7}$

Step 5.3.9.2

Cancel the common factor.

$f (- \frac{7 x - 3}{7 - 6 x}) = \frac{x (- 6 x + 7)}{- 6 x + 7}$

Step 5.3.9.3

Divide $x$ by $1$ .

$f (- \frac{7 x - 3}{7 - 6 x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x - 3}{7 - 6 x}$ is the inverse of $f (x) = \frac{7 x - 3}{6 x - 7}$ .

$f^{- 1} (x) = - \frac{7 x - 3}{7 - 6 x}$